64c3814
==== TABLE OF CONTENTS ====
Irrational-cubic-fourfolds-with-geometric-K3-categories//bounds.tex
Irrational-cubic-fourfolds-with-geometric-K3-categories//cubics.tex
Irrational-cubic-fourfolds-with-geometric-K3-categories//gw.tex
Irrational-cubic-fourfolds-with-geometric-K3-categories//introduction.tex
Irrational-cubic-fourfolds-with-geometric-K3-categories//paper.tex
Irrational-cubic-fourfolds-with-geometric-K3-categories//preliminaries.tex
Irrational-cubic-fourfolds-with-geometric-K3-categories//subfields.tex
Irrational-cubic-fourfolds-with-geometric-K3-categories//surface-links.tex
Irrational-cubic-fourfolds-with-geometric-K3-categories//references.bib
===========================
==== BEGIN Irrational-cubic-fourfolds-with-geometric-K3-categories//bounds.tex ====
\section{Uniform bounds for the link diagrams}\label{bd:section}
This section supplies the uniform bounds used in the Sarkisov argument.
Theorem~\ref{bd:exclusions} converts bounded varieties and marked covers
into degree bounds for K3 tests. We then construct a central model at
each Sarkisov wall, preserving a fixed discrepancy bound, and recover
the link as contractions of that model. Theorem~\ref{bd:diagrams}
bounds these diagrams on geometric generic fibres, including the maps
and marked bad loci that control the branch locations of later covers.
Theorem~\ref{bd:terminalization} separately bounds models extracting
only low-discrepancy valuations from a bounded log-Fano pair; this will
control a relative minimal model without bounding its resolving start.
Finally, Corollary~\ref{bd:choice-order} first fixes
one ramification coefficient from the ordinary fourfold diagrams;
only then do we bound the auxiliary log diagrams and choose the final
K3 degree threshold.
\subsection{Elementary exclusions for K3 fields}
In this subsection a \emph{degree-$d$ K3 surface} means a smooth projective
K3 surface $S$ with $\operatorname{Pic}(S)=\Z L$, where $L$ is ample and
$L^2=d$. All the bounds below are independent of the choice of $S$, and
therefore also of any further restriction on its transcendental Hodge
structure or automorphism group.
\begin{lemma}[Pencils on a K3 surface]\label{bd:pencil}
Let $S$ be a degree-$d$ K3 surface. If a dominant rational map from $S$
to a smooth projective curve has connected general fibre, that curve is
$\mathbb P^1$. If $g$ is the genus of the smooth general fibre on a
resolution of the map, then
\[
2g-2\geq\sqrt d.
\]
For an arbitrary dominant rational map to a curve the same inequality
holds for every geometric general fibre component, after normalization.
\end{lemma}
\begin{proof}
Resolve the map by point blow-ups $\pi:Y\to S$. Since $h^1(Y,\mathcal
O_Y)=0$, pullback of differentials shows that the base of its Stein
factorization has genus zero. Thus, after Stein factorization, we have
a morphism $f:Y\to\mathbb P^1$ with smooth connected general fibre $F$.
The images on $S$ of these fibres form a pencil without fixed components,
of class $nL$ for some integer $n\geq1$. In the total-transform basis of
the successive exceptional curves write
\[
F=\pi^*(nL)-\sum_i m_iE_i^*,\qquad
K_Y=\sum_iE_i^*,\qquad m_i\geq0.
\]
The exceptional basis is orthogonal, with $(E_i^*)^2=-1$. Consequently
adjunction and $F^2=0$ give
\[
\sum_i m_i^2=n^2d,\qquad 2g-2=K_Y\cdot F=\sum_i m_i.
\]
It follows that $2g-2\geq(\sum_i m_i^2)^{1/2}=n\sqrt d$.
Stein factorization accounts for the geometric components in the last
assertion.
\end{proof}
We use the following precise boundedness conventions. A collection of
projective varieties is \emph{bounded} if its members occur as geometric
fibres of finitely many projective morphisms of schemes of finite type.
It is \emph{birationally bounded} if each member is birational to such a
fibre. For pairs $(Y,D)$, boundedness includes the reduced proper closed
subset $D\subsetneq Y$: it must be supplied by a closed subset of the
same family. A finite cover of $(Y,D)$ means the normalization of $Y$
in a finite field extension, required to be \emph{\'etale over $Y\setminus
D$}. Thus the boundary hypothesis controls branch locations, not just
the degree of the extension.
Here and below one can choose a bounded collection of smooth
projective birational models of the irreducible components of a bounded
collection. To justify this
operation, work on an irreducible parameter space, separate its geometric
generic components by a finite extension of the function field, resolve
them, and spread the resolutions. After shrinking, the spread models
are smooth and the maps are birational on the appropriate fibre
components. Apply the same construction to the remaining closed subset.
Noetherian induction terminates this procedure. The same argument works
with a marked closed subset, using embedded resolution and including the
exceptional locus in the new boundary. It also proves that irreducible
components of fibres of a fixed projective family, and smooth projective
models of those components, are birationally bounded. These operations
allow finite base changes of parameter spaces; they impose no bound on a
particular map from one member to another.
\begin{lemma}[Bounded threefolds and their K3 quotients]
\label{bd:bounded-mrc}
For a birationally bounded collection of varieties of dimension at most
three, there is an integer $b$ such that any degree-$d$ K3 surface
birational to the MRC base of a member satisfies $d\leq b$.
\end{lemma}
\begin{proof}
First consider a bounded smooth projective surface $Y$ birational to
$S$. Its minimal-model morphism $\mu:Y\to S$ is a succession of point
blow-downs. Let $A$ be a very ample divisor from the bounded family.
In the total-transform exceptional basis,
\[
A=\mu^*B-\sum_i a_iE_i^*,\qquad
K_Y=\sum_iE_i^*,\qquad a_i=A\cdot E_i^*>0.
\]
The pushforward $B$ is ample: at each blow-down its square increases and
its intersection with every curve is positive, so the surface
Nakai--Moishezon criterion applies \cite[Theorem~1]{Cartier66}. It is an integral multiple of $L$.
Hence
\begin{equation}\label{bd:surface-degree}
d\leq B^2=A^2+\sum_i a_i^2
\leq A^2+(A\cdot K_Y)^2.
\end{equation}
The right-hand side takes only finitely many values in a bounded smooth
projective family after a finite stratification. This proves the
assertion in dimension two; in smaller dimensions a K3 MRC base is
impossible.
It remains to bound the birational types of the K3 bases of bounded
smooth projective threefolds. Put these threefolds into finitely many
smooth projective families $\mathcal X\to T$, with relatively very ample
divisor $H$. We use the fixed-polynomial Hilbert space and its open
graph locus \cite[Tags~0DPH, 0D1A and~0D1B]{StacksProject057}, together with
semicontinuity and base change \cite[Tags~0BDN and~0B91]{StacksProject057}.
For $e\geq1$, the free locus in the relative scheme of
degree-$e$ maps $\mathbb P^1\to\mathcal X/T$ is open. Its morphism to
$T$ is smooth: for a free map $a$ the obstruction space
$H^1(\mathbb P^1,a^*T_{\mathcal X_t})$ vanishes. Its image is therefore
open. If $U_D$ is the union of these images for $1\leq e\leq D$, then
the $U_D$ are ascending open subsets of the noetherian space $T$.
Their union is precisely the locus of uniruled geometric fibres, since
in characteristic zero a smooth projective variety is uniruled if and
only if it has a nonconstant free rational curve. The union is
quasi-compact, so $U_D$ equals that union for some $D$.
These deformation statements follow, for example, from
\cite[Remark~9, Proposition~10 and Theorem~15]{AraujoKollar02}; this argument proves
the uniformity rather than assuming that the uniruled fibres themselves
form a separately specified family.
Suppose now that the MRC quotient of $X=\mathcal X_t$ is a surface
birational to $S$. Use its almost-holomorphic presentation
$r:U\to S^\circ$ with proper smooth rationally connected fibres;
see \cite[Chapter~IV, Section~5]{Kollar96}.
Its fibres are smooth copies of $\mathbb P^1$. A degree-at-most-$D$
free map on $X$ deforms in a covering family. A very general member
meets a very general fibre of $r$ and is contained in it, by the MRC
property. Since that fibre is a complete integral curve, the image of
the map is the whole fibre. The $H$-degree $e$ of the fibre is thus at
most $D$.
For such a fibre $F$, smoothness of $r$ gives
$N_{F/X}\simeq\mathcal O_{\mathbb P^1}^{\oplus2}$. Its Hilbert scheme
is therefore smooth of dimension two at $[F]$. The family of fibres
defines an injective map from $S^\circ$ to this Hilbert scheme, with
isomorphic tangent spaces; its closure is an irreducible component of
$\operatorname{Hilb}^{en+1}(X)$ birational to $S$. The components so
obtained are components of fibres of the finite-type projective scheme
\[
\coprod_{e=1}^{D}
\operatorname{Hilb}^{en+1}(\mathcal X/T).
\]
They have bounded smooth projective birational models by the preceding
bounded-family reduction. Apply \eqref{bd:surface-degree} to these
surfaces. This proves a uniform bound independent of the threefold and
of its MRC quotient map.
\end{proof}
\begin{theorem}[Bounded data exclude large K3 degrees]
\label{bd:exclusions}
Fix the finite-type families occurring in each of the following
statements, and fix an integer $N\geq1$. There is an integer $b$,
depending only on these families and on $N$, with the following
properties.
\begin{enumerate}[label=\textup{(\roman*)}]
\item No member of a specified birationally bounded collection of
varieties of dimension at most three has a degree-$d$ K3 MRC base with
$d>b$.
\item Let $(Y,D)$ range over a bounded collection of projective curve
or surface pairs. The smooth projective curves covering the curve
members with degree at most $N$, \emph{\'etale off $D$}, have bounded
genus. No surface field obtained by such a cover of a surface member is
the field of a degree-$d$ K3 surface for $d>b$.
\item Let $T\dashrightarrow C$ be a dominant map from an integral
complex surface to a smooth projective curve. Suppose each normalized
geometric generic fibre component is a degree-at-most-$N$ cover of a
member of a fixed bounded collection of curve pairs, \emph{\'etale off
the marked boundary}. Then $T$ is not birational to a degree-$d$ K3
surface for $d>b$.
\item Let $T\dashrightarrow C$ be a dominant map from an integral
complex threefold to a smooth projective curve. Suppose the geometric
generic fibre components are uniruled and belong to a fixed
birationally bounded collection of surfaces. Then the MRC base of
$T$ is not a degree-$d$ K3 surface for $d>b$.
\end{enumerate}
The bounded geometric data in \textup{(iii)} and \textup{(iv)} are
understood after algebraic closure of $\C(C)$; neither the total spaces
$T$ nor the maps to $C$ are required to be bounded. The assertions
apply to each field factor of a finite algebra separately.
\end{theorem}
\begin{proof}
Part~\textup{(i)} is Lemma~\ref{bd:bounded-mrc}.
For \textup{(ii)}, pass to bounded smooth projective models of the pairs,
including the exceptional locus in the boundary. The normalized
base-changed covers are still \emph{\'etale} on the complementary open.
For a smooth curve $A$ of genus $g_A$ with at most $r$ marked points,
and a connected degree-$e$ cover $A'\to A$ unramified elsewhere,
Riemann--Hurwitz gives
\begin{equation}\label{bd:hurwitz}
2g(A')-2\leq e(2g_A-2)+(e-1)r.
\end{equation}
Indeed, over any one branch point the ramification contribution is
$e$ minus the number of points above it, and is at most $e-1$.
This proves the curve assertion uniformly for $e\leq N$.
For a bounded smooth surface pair choose a pencil of hyperplane sections
for a relatively very ample divisor $H$. A general section $A$ is smooth,
has uniformly bounded genus by adjunction, and meets the divisorial part
of $D$ in uniformly boundedly many points. Choose it to avoid the
zero-dimensional part of $D$. For any one cover, general members can
also avoid the images of the singular points of its normalization;
normal surfaces have only finitely many singular points. Each
normalized component over $A$ has degree at most $N$ and is ramified
only over $A\cap D$, so \eqref{bd:hurwitz} gives a uniform genus bound
$G$. Resolve the pulled-back pencil on a smooth model of the covering
surface and take its Stein factorization. Its smooth connected general
fibres are precisely these normalized components. If the surface field
were that of $S$, Lemma~\ref{bd:pencil} would give
$\sqrt d\leq2G-2$. This proves the surface assertion. The pencil can
depend on the individual cover; only its genus and boundary-intersection
bounds need to be uniform.
The same curve estimate over $\overline{\C(C)}$ bounds the genera of
the geometric generic components in \textup{(iii)}. If $T$ were
birational to $S$, the given map to $C$, after resolution and Stein
factorization, would give a pencil on $S$. Lemma~\ref{bd:pencil}
again bounds $d$. No genus bound for the parameter curve $C$ is needed.
For \textup{(iv)}, suppose the MRC base of $T$ is birational to $S$,
and write $r:T\dashrightarrow S$ and $f:T\dashrightarrow C$.
Put $F=\C(C)$ and choose an algebraic closure $\overline F$. Resolve
the maps when necessary. If $f$ were nonconstant on a general fibre
of $r$, the map $(r,f):T\dashrightarrow S\times C$ would be dominant
and generically finite. A component of a geometric generic surface
fibre of $f$ would then dominate $S_{\overline F}$ generically finitely. Such a
component is uniruled, whereas a generically finite dominant image of
an uniruled surface is uniruled. This contradicts the fact that a K3
surface is not uniruled in characteristic zero.
Consequently $f=h\circ r$ for a dominant rational map $h:S\dashrightarrow
C$. This is descent along the geometrically integral general fibres
of the MRC quotient, or equivalently the relative algebraic closedness
of $\C(S)$ in $\C(T)$. Resolve $h$ and take its Stein factorization.
Over $\overline F$, let $\Gamma$ be a smooth geometric generic fibre
of the resulting connected-fibre pencil, of genus $g$. A geometric
generic component $P$ of $f$ dominates the corresponding $\Gamma$.
For a smooth projective model $\widetilde P$ over $\overline F$ this implies
\[
g\leq h^0(\widetilde P,\Omega^1_{\widetilde P})
=h^1(\widetilde P,\mathcal O_{\widetilde P}).
\]
To see the inequality, resolve the rational map to $\Gamma$ and pull
back regular one-forms; pullback is injective for a dominant map in
characteristic zero, and irregularity is a birational invariant of
smooth projective surfaces. The irregularities of $\widetilde P$ are
uniformly bounded by bounded smooth projective birational models and
upper semicontinuity. Lemma~\ref{bd:pencil} therefore bounds $d$.
All genus, intersection, and coherent-cohomology bounds used here are
preserved by extension of algebraically closed fields. The same proofs
therefore apply to the geometric generic data specified in
\textup{(iii)} and \textup{(iv)}. Taking the maximum of the finitely many
bounds proves all the assertions.
\end{proof}
\subsection{Central models for the two Sarkisov programs}
The boundedness problem starts with a wall in a Sarkisov factorization.
We first construct a central model that retains the prime valuations of
the link and a fixed lower bound for log discrepancies. Its geometric
generic fibre will supply the log-Fano pair to which boundedness is
applied. The next subsection will then recover the contraction maps and
their marked loci from these pairs.
For a pair $(Y,B)$ we write $a(F,Y,B)$ for \emph{log discrepancy}; a
prime divisor on $Y$ of coefficient $b$ has log discrepancy $1-b$.
A pair is $\epsilon$-lc if every log discrepancy is at least
$\epsilon$. Terminality concerns exceptional divisors and means
$a(F,Y,B)>1$ for such divisors. In particular, terminality of the
underlying variety is used for the ordinary Sarkisov program, whereas
only a fixed positive lower bound for log discrepancies is needed for
the auxiliary log program.
A variety is of \emph{Fano type} if it admits an effective rational
boundary making it klt log Fano. We use the relative version over a
base as well. By finite generation, a $\Q$-factorial variety of Fano
type is a Mori dream space: its movable cone has finitely many
semiample chambers, and an MMP for any divisor terminates
\cite[Corollary~1.3.2]{BCHM10}. The chamber description includes
the contraction associated with each face
\cite[Proposition~1.11]{HuKeel00}.
The contraction and relative basepoint-free statements we use are
\cite[Theorem~4.5.2 and Corollary~6.9.4]{FujinoFoundationsDraft057}.
For the sign and uniqueness of exceptional differences we use the
negativity lemma \cite[Lemma~11.60]{KollarFamiliesDraft057}.
A \emph{contraction morphism} is a projective surjective morphism
$f:Y\to Z$ of normal varieties with $f_*\mathcal O_Y=\mathcal O_Z$.
A \emph{birational contraction} is a birational map whose inverse
contracts no divisor.
A \emph{small modification} is a birational map that is an isomorphism
in codimension one. A \emph{small $\Q$-factorialization} is a projective
small birational morphism with $\Q$-factorial source.
For a $\Q$-factorial Fano-type source $Y$, a
\emph{rational contraction} means a composite
$Y\dashrightarrow Y'\to Z$, where the first map is a small
$\Q$-factorial modification and the second is a contraction morphism.
The target $Z$ need only be normal and projective. We use the same
definition over an indicated base, with both maps over that base.
For an integral Weil divisor $D$ on a normal projective variety $Y$,
write
\[
R(Y,D)=\bigoplus_{m\geq0}H^0(Y,\mathcal O_Y(mD)),
\]
where $\mathcal O_Y(mD)$ is the divisorial reflexive sheaf. For a
rational divisor we use a sufficiently divisible integral multiple;
further Veronese subrings do not change $\operatorname{Proj}$. In
the finitely generated cases used below, the induced rational map
$Y\dashrightarrow\operatorname{Proj}R(Y,D)$ is its \emph{ample-model map}.
We use the ample-model formulation of the Sarkisov theorem
\cite[Theorem~1.3, Theorem~3.3, Theorem~3.7, and Lemma~4.1]{HM13}.
Start with two Mori fibre-space outcomes of one klt $K_W+\Phi$
program on a smooth projective variety $W$. The theorem connects
them using a finite-dimensional family of klt boundaries $\Theta$
with $\Theta-\Phi$ ample on $W$; this family and its ample models
are the \emph{geography} used below.
All paths whose diagrams we bound below are chosen from this geography.
At a linking wall, let $N_1,N_2$ be the $\Q$-factorial middle models
on the two sides. The local diagram supplied by the two-ray construction is
\[
\begin{tikzcd}[row sep=small,column sep=large]
N_1 \arrow[rr,dashed,"\text{small over }Q"] \arrow[d,"e_1"]
&& N_2 \arrow[d,"e_2"] \\
V_1 \arrow[d,"\pi_1"] && V_2 \arrow[d,"\pi_2"] \\
B_1 \arrow[dr,"q_1"'] && B_2 \arrow[dl,"q_2"] \\
& Q &
\end{tikzcd}
\]
Each $N_i\to Q$ has relative Picard rank two. The arrow $e_i$ is
either the identity or an elementary divisorial contraction;
$\pi_i$ is the end Mori fibre contraction, of relative rank one.
Additivity of the relative ranks in these morphism towers gives
\begin{equation}\label{bd:rank-tower}
2=\rho(N_i/Q)=\rho(N_i/V_i)+1+\rho(B_i/Q).
\end{equation}
These are ranks over the original common base $Q$, before any
extension of its function field. The arrows $q_i$ may be small
contractions. All maps $W\dashrightarrow N_i$ are birational
contractions, and the transforms of $K_W+\Theta$ on the $N_i$ are
semiample and pulled back from $Q$. Thus the middle small models
share their prime valuations; the divisorial exits $e_i$ can remove
valuations.
The uniformity we need is a deduction from this construction,
not an assumption that arbitrary Fano-type varieties are bounded.
A relative negative program over a projective target is also a
negative program in the absolute category used here. Indeed its
cone of contracted curves is the face cut out by the pullback of
an ample divisor on that target; a negative extremal ray of this
face is extremal in the full cone. The contractions and flips
remain projective. Thus a program run first over one projective
target and then over another, for example over $T$ and then over
$Q$, is an allowed common-start program. Its final relative
rank-one fibre contraction is a Mori fibre space in this category.
\begin{lemma}[A bounded-discrepancy central model]
\label{bd:central}
Consider a wall in this geography on a smooth projective start
$(W,\Phi)$. Assume that $(W,\Phi)$ is $\epsilon$-lc. Then the
relative small models in the link have a common set of prime
valuations and admit a small modification $N^{\mathrm c}/Q$
such that
\[
-(K_{N^{\mathrm c}}+\Phi_{N^{\mathrm c}})
\quad\text{is nef and big over }Q.
\]
Its log-anticanonical ample-model map
$N^{\mathrm c}\to Z^{\mathrm c}$ is small. Both
$N^{\mathrm c}$ and $Z^{\mathrm c}$ are $\epsilon$-lc with
their transformed boundaries. All the link vertices and
contraction bases are rational contractions of a small
$\Q$-factorialization of $Z^{\mathrm c}$. If $\Phi=0$, all
the small models can be taken terminal.
\end{lemma}
\begin{proof}
Take $N=N_1$ in the displayed wall diagram, let
$A=\Theta-\Phi$ be the ample difference on $W$, and write
$A_N,\Theta_N,\Phi_N$ for transforms. We first transfer the
discrepancy bound from $W$ to $N$. We then make the original
log-anticanonical class nef and pass to its ample model. That last
map must be small: a divisorial contraction would lose one of the
prime valuations we need to retain.
On a graph resolution
with projections $p$ to $W$ and $q$ to $N$,
\begin{align}
p^*(K_W+\Theta)-q^*(K_N+\Theta_N)&=E_\Theta\geq0,
\label{bd:wall-comparison}\\
p^*A-q^*A_N&\leq0.\label{bd:ample-comparison}
\end{align}
For this comparison at the wall, approach $\Theta$ by interior parameters
$\Theta_i$ in the chamber defining $N$.
By \cite[Theorem~3.3(3)]{HM13}, $W\dashrightarrow N$ is a log terminal
model for $K_W+\Theta_i$, so the exceptional difference on the same graph
resolution is effective. Its finitely many coefficients depend linearly
on the boundary. Taking their limits gives $E_\Theta\geq0$;
\cite[Theorem~3.3(2)]{HM13} supplies the contraction $N\to Q$.
Thus the first difference is exceptional and nonnegative at the wall. The second is also $q$-exceptional,
since the map does not extract. On a $q$-contracted curve its
degree equals that of $p^*A$, so it is $q$-nef; the negativity
lemma proves \eqref{bd:ample-comparison}. Subtracting gives
\begin{equation}\label{bd:original-comparison}
p^*(K_W+\Phi)-q^*(K_N+\Phi_N)\geq0.
\end{equation}
It follows valuation by valuation that the original
$\epsilon$-lc bound is retained on $N$.
There is a useful extension of this argument. The divisor
$K_N+\Theta_N$ is pulled back from $Q$. Any small
$\Q$-factorial modification $N'\to Q$ has the same
codimension-one divisor and the same pullback relation.
Thus $N\dashrightarrow N'$ is crepant for $K_N+\Theta_N$.
The comparison \eqref{bd:wall-comparison} remains nonnegative
for $W\dashrightarrow N'$, and the ample subtraction proves
\eqref{bd:original-comparison} for $N'$ as well. In particular,
the bound holds on any required relative small chamber; there
is no need to locate all those chambers in one chosen
two-dimensional slice.
The pair $(N,\Theta_N)$ is klt, its boundary is big over $Q$,
and its log-canonical divisor is trivial over $Q$. Writing
that boundary, up to relative rational linear equivalence, as
an ample divisor plus an effective divisor, a sufficiently
small perturbation makes a klt log-Fano boundary. Therefore
$N/Q$ is of Fano type. Furthermore,
\[
-(K_N+\Phi_N)\equiv_Q A_N.
\]
The transform of an ample class by a birational contraction is
in the interior of the movable cone: choose divisor classes
on $W$ mapping onto a basis on $N$, perturb $A$ slightly in
both directions in this basis, and push their ample systems
forward. No prime divisor is fixed, because none is extracted.
The same argument works after passing to the relative numerical
space over $Q$.
Finite generation now makes this movable class nef on a small
model $N^{\mathrm c}/Q$. Only small steps are needed: a
negative divisorial contraction would force its exceptional
divisor to be fixed in the corresponding system. The class
is big, and its semiample contraction cannot be divisorial
because it lies in the interior of the movable cone. Indeed
a nef class vanishing on curves covering a contracted divisor
lies on the boundary of the movable cone, as follows by
intersecting those curves with effective divisors avoiding
that divisor. Its ample-model map $N^{\mathrm c}\to Z^{\mathrm c}$
is therefore small.
The log-canonical divisor is crepant for this last contraction,
so the discrepancy bound passes to $Z^{\mathrm c}$. Its small
$\Q$-factorializations have precisely the common prime
valuations of the link. In particular, $N^{\mathrm c}\to Z^{\mathrm c}$
is one such small $\Q$-factorialization, and
$N^{\mathrm c}\dashrightarrow N_i$ is small for $i=1,2$.
Composing with the contraction towers
$N_i\to V_i\to B_i\to Q$ in the wall diagram gives exactly
the rational contractions asserted in the statement.
When $\Phi=0$, exceptional valuations above $W$ already have
log discrepancy greater than one. A divisor of $W$ that is
contracted has a strictly negative coefficient in
\eqref{bd:ample-comparison}: intersect with a general curve
in its positive-dimensional contracted fibre, on which $A$
has positive degree. Its coefficient in
\eqref{bd:original-comparison} is consequently strictly
positive. This gives terminality also for these remaining
exceptional valuations. The same reasoning applies to each
small model as above.
\end{proof}
To compare higher smooth starts, we use the following SNC discrepancy
estimate. Let $(R,\Gamma)$ be a smooth pair with SNC support and rational
coefficients $b_i<1$, allowing negative coefficients, and let $F$ be an
exceptional divisorial valuation. At the generic point of its center of
codimension $r$, choose regular parameters $x_1,\ldots,x_r$ containing
the equations of the boundary components through that center. Then
\begin{equation}\label{bd:snc-discrepancy}
a(F,R,\Gamma)\geq
\sum_{i=1}^r(1-b_i)\operatorname{ord}_F(x_i),
\end{equation}
where $b_i=0$ for parameters not defining a boundary component.
This follows from the Jacobian formula for a divisorial valuation
over a regular local ring, followed by subtracting the boundary
orders. All orders are positive integers.
\begin{lemma}[Higher common starts preserve the original marks]
\label{bd:higher-start}
Let $(W,\Phi)$ be smooth projective with SNC boundary whose
coefficients belong to $\{0,c\}$, where $0<c<1/4$ and
$\dim W\leq3$. Suppose two Mori fibre spaces are outputs of
negative $(K_W+\Phi)$ programs. On a higher smooth common
start resolving the maps to both endpoints, keep the marks on
original prime divisors and assign
coefficient either $0$ or $c$ to any new prime divisor, with
SNC total support. The same two outputs remain outcomes of
the program for this new marked pair. In its geography the
original marked discrepancy bound can be taken to be
$\epsilon=1-c$, independently of the higher start.
The analogous assertion holds for ordinary terminal Mori
fibre spaces with zero boundary and $\epsilon=1$, including
the four-dimensional paths used below.
\end{lemma}
\begin{proof}
Write $(W',\Phi')$ for the higher smooth start with its new marking.
On the original smooth SNC pair, every exceptional valuation
has log discrepancy greater than one: \eqref{bd:snc-discrepancy}
gives the bound $2(1-c)>1$ for rational $c$. For irrational $c$,
apply it to rational coefficients decreasing to $c$ and use the
affine dependence of log discrepancy on the boundary coefficients.
The non-exceptional divisors
have discrepancy at least $1-c$. These facts also hold on
the new smooth SNC start directly, without comparing the
number of its exceptional divisors.
Fix one endpoint $V$ and write $f:W'\to V$ for the resolved map.
For a divisor of $W$ contracted by
its negative program, discrepancy strictly improves. If
its retained marking is $b\in\{0,c\}$, then on the higher
start its coefficient in the difference from the pullback
of $K_V+\Phi_V$ is
\[
a(F,V,\Phi_V)-1+b>0.
\]
For a divisor exceptional over $W$, monotonicity gives
$a(F,V,\Phi_V)\geq a(F,W,\Phi)>1$, and adding a new mark
$b\geq0$ again makes this coefficient positive. Thus the
entire difference over $V$ is effective with strictly
positive coefficients on every endpoint-exceptional divisor.
Run the relative program over $V$. Existence follows from
relative bigness for a birational morphism
\cite[Theorem~1.2]{BCHM10}. At a relative minimal model the
exceptional difference is nef over $V$ and hence nonpositive
by negativity. Since all the above positive exceptional
coefficients must have been contracted, the resulting map
to $V$ is small. A projective small birational morphism to
a $\Q$-factorial variety is an isomorphism. This realizes
the same endpoint, after which its Mori contraction can be
used. The argument works for both endpoints.
We spell out the endpoint perturbations from
\cite[Lemma~4.1]{HM13}. Write $f_i:W'\to V_i$ for the two resolved
maps and $\pi_i:V_i\to B_i$ for their Mori contractions.
Choose small general effective ample rational
divisors $A,H_1,\ldots,H_s$, with the classes of the $H_j$ spanning
$N^1(W')$, and put $H=A+\sum_j H_j$. If $c$ is irrational, also
choose a rational boundary $\Phi_0\leq\Phi'$ sufficiently close to
$\Phi'$; for rational $c$ take $\Phi_0=\Phi'$. Make these choices so
that $A+\Phi_0-\Phi'$ is ample, both $f_i$ are
$(K_{W'}+\Phi_0+H)$-negative, and
$-(K_{V_i}+(f_i)_*\Phi_0+(f_i)_*H)$ is $\pi_i$-ample.
The exceptional differences have finitely many strictly positive
coefficients, which stay positive under these small changes;
the ampleness conditions are open as well.
For a sufficiently positive ample rational
divisor $C_i$ on $B_i$, the divisor
\[
L_i=-(K_{V_i}+(f_i)_*\Phi_0+(f_i)_*H)+\pi_i^*C_i
\]
is ample. Choose a general effective rational representative
$F_i^{V_i}\sim_{\Q}L_i$ and put $F_i=f_i^*F_i^{V_i}$. Then
\[
\Theta_i=\Phi_0+H+F_i,\qquad
K_{V_i}+(f_i)_*\Theta_i\sim_{\Q}\pi_i^*C_i.
\]
Here $\Theta_i-\Phi'$ is ample: it is the sum of the ample divisor
$A+\Phi_0-\Phi'$ and the nef divisors $H_j,F_i$. Simultaneous general
representatives of sufficiently divisible multiples make the
dominating pair $(W',\Phi'+H+F_1+F_2)$ klt.
The negativity of $f_i$ therefore makes $B_i$ the required endpoint
ample model. The spanning classes $H_j$ and general two-dimensional
slice in \cite[Lemma~4.1]{HM13} complete these two perturbations to
one geography, retaining $\Phi'$ as the fixed original boundary.
Lemma~\ref{bd:central} now uses the unchanged lower bound
$1-c$. With zero boundary the same proof starts from the
strictly positive discrepancies of terminal endpoints.
\end{proof}
\subsection{Bounded marked pairs and their contraction diagrams}
The central models preserve the discrepancy bound independently of the
chosen start. To bound the resulting links, we must control more than
these varieties: the contraction maps and their marked bad loci determine
the branch locations of the finite covers used later. We now prove this
upgrade for bounded log-Fano pairs. Theorem~\ref{bd:diagrams} will combine
it with Birkar's boundedness theorem for the central models.
In a \emph{marked contraction diagram}, the marks are finitely many
closed subsets, including specified divisors, and the arrows are
birational contractions or contraction morphisms. Diagrams
are considered up to compatible isomorphisms of their vertices; the
birational arrows use the common identification of function fields.
More precisely, every vertex field is embedded in the field of the
initial variety and every arrow respects these embeddings; in
particular, a cycle of birational arrows induces the identity on
that field.
Boundedness requires families for the vertices, marks, and graphs of
the arrows. In the contraction diagrams constructed below, the arrows
are the ample-model maps with these field identifications.
For a rational contraction we record both its factorization arrows
and the graph of their composite.
Graph resolutions will be chosen from the resulting
finite-type families. Repetition of a vertex or arrow does not add new data.
We first address the family issue that occurs when boundedness is
initially given without a relative $\Q$-Cartier divisor.
\begin{lemma}[Stratification of the marked log-Fano data]
\label{bd:cartier-strata}
Let $\mathcal P$ be a bounded collection of marked projective klt
log-Fano pairs $(Y,B)$, with coefficients of $B$ in a fixed finite
subset of $\Q$. Finitely many stratified families, after finite
base changes of their parameter spaces, can be chosen so that:
\begin{enumerate}[label=\textup{(\roman*)}]
\item the log-canonical divisor is relatively $\Q$-Cartier, with a
common Cartier multiple on each stratum;
\item there is a simultaneous log resolution of the marked support,
and the coefficients of the crepant pullback of $B$ are constant on
each stratum;
\item any further specified rational Weil divisor, with bounded
marked support and coefficients in a fixed finite subset of $\Q$,
which is $\Q$-Cartier for every member has a common Cartier multiple
on each stratum, and its exceptional pullback coefficients are constant.
\end{enumerate}
The same assertions about $\Q$-Cartier classes hold for a bounded
collection of varieties with rational singularities whose smooth
projective resolutions have $H^1(\mathcal O)=H^2(\mathcal O)=0$.
In these families $\Q$-factoriality is a constructible condition.
\end{lemma}
\begin{proof}
We give the generic argument, and then apply noetherian induction.
Take the closure of the parameter points under consideration and an
irreducible component of this closure. After shrinking and a
generically finite base change, we have flat normal geometric fibres
$\mathcal Y_t$, a family $\pi:\mathcal R\to\mathcal Y$ of resolutions,
and smooth projective fibres $\mathcal R_t$ on which the marked strict
transforms and exceptional divisors form a relative SNC divisor.
The individual exceptional components can also be separated by this
base change. These are the usual spreading operations on a resolution
of the geometric generic fibre.
For the members in question, rational singularities and vanishing
give
\[
H^i(\mathcal Y_t,\mathcal O_{\mathcal Y_t})
=H^i(\mathcal R_t,\mathcal O_{\mathcal R_t})=0\quad(i=1,2).
\]
For log-Fano pairs, the downstairs vanishing follows from
Kawamata--Viehweg vanishing and the equality follows from rational
singularities; see
\cite[Theorem~3.13.1 and Corollary~5.7.7]{FujinoFoundationsDraft057}.
The parameter points of these members are Zariski dense in the
closure chosen above. Thus they satisfy the dense-subset hypothesis
of the relative Picard-space theorem \cite[Theorem~1.3]{CLZ26}:
rational singularities and $H^1(\mathcal O)=H^2(\mathcal O)=0$
are required on that dense subset. Upper semicontinuity also gives
the displayed vanishings on an open set. The theorem permits a further generically finite
base change and shrinking such that the rational numerical divisor
spaces of $\mathcal Y_t$ and $\mathcal R_t$, their pullback map, and
the exceptional divisor classes are identified with fixed rational
linear data. In particular, choose relative Cartier divisors whose
classes span the downstairs space on every fibre.
Here is the required $\Q$-Cartier criterion. For a rational Weil
divisor $D$ on a fibre, let $\widetilde D$ be its strict transform.
Then
\begin{equation}\label{bd:cartier-test}
D\text{ is }\Q\text{-Cartier}
\quad\Longleftrightarrow\quad
[\widetilde D]\in
\pi^*N^1(\mathcal Y_t)_{\Q}
+\sum_E\Q[E]\ \subset N^1(\mathcal R_t)_{\Q}.
\end{equation}
The forward implication follows by pullback. For the converse,
numerical equivalence of rational divisors on $\mathcal R_t$ is
rational linear equivalence: its $\operatorname{Pic}^0$ vanishes,
and the numerically trivial part of the N\'eron--Severi group is
torsion. Thus the asserted relation, after clearing denominators,
is a linear-equivalence relation with a Cartier divisor pulled back
from $\mathcal Y_t$ and exceptional divisors. Pushing down proves
that a multiple of $D$ is linearly equivalent to a Cartier divisor.
This is precisely the converse in \eqref{bd:cartier-test}.
Apply the criterion to the fixed rational classes on the resolution.
For the log-canonical class one may use
$K_{\mathcal R_t}+\widetilde B_t$ modulo exceptional classes; this
does not presuppose that $K_{\mathcal Y_t}+B_t$ is $\Q$-Cartier.
The membership in \eqref{bd:cartier-test} holds on the dense set of
members under consideration, and the linear data are fixed. It
therefore holds at the geometric generic point, with fixed rational
coefficients in a basis of relative Cartier classes. Rational linear
equivalence upstairs, followed by pushdown, proves that the generic
log-canonical divisor is $\Q$-Cartier. Clear denominators and spread
this linear equivalence. Its possible vertical errors disappear
after shrinking, giving a relative Cartier multiple. This argument
applies simultaneously to each additional specified class.
Exceptional coefficients in a pullback are unique. Indeed, an
exceptional rational divisor numerically trivial over its target is
zero, by applying the negativity lemma to it and its negative.
They are thus the solutions of fixed rational linear equations on
the resolution, and are constant after shrinking. The klt condition
is now the finite set of strict inequalities saying that the crepant
SNC coefficients are less than one. Relative log-anticanonical
ampleness is open. Thus an open set consists entirely of the required
log-Fano pairs, with all the asserted properties.
Finally, all Weil divisors downstairs are $\Q$-Cartier exactly when
the right side of \eqref{bd:cartier-test} exhausts
$N^1(\mathcal R_t)_{\Q}$: every divisor upstairs pushes to a Weil
divisor, and the same argument applies. This is again a condition
on the fixed linear data. Applying the construction to the
remaining closed parameter sets proves all the assertions by
noetherian induction.
\end{proof}
Choi--Li--Zhou prove boundedness of the birational contraction models
of members of a fixed bounded family of Fano-type varieties
\cite[Theorem~1.7 and Section~5.1]{CLZ26}.
We use their fibrewise small $\Q$-factorializations and the comparison
between relative and fibre MMPs
\cite[Theorems~1.4--1.6 and Lemma~5.5]{CLZ26}.
The additional data in the following statement are the marked supports,
graphs, exceptional loci and their resolutions.
The proof uses finite generation to list the canonical contraction
maps on a generic member, the cited family results to retain the
complete list after shrinking, and then spreading to include their
graphs and marks.
\begin{lemma}[Uniform contraction diagrams]
\label{bd:contractions}
For a bounded collection of projective klt log-Fano pairs with fixed
finite coefficient set, all diagrams obtained from a small
$\Q$-factorialization by small modifications and rational contractions
are bounded. The assertion includes the transforms and images of a
bounded marked support, the exceptional loci of the arrows, and their
chosen graph resolutions. It also includes a non-extracting birational map
followed by a contraction morphism.
Here a diagram is a subdiagram of the collection of contraction models
and their canonical arrows compatible with the fixed function field.
\end{lemma}
\begin{proof}
Lemma~\ref{bd:cartier-strata} supplies families of Fano-type fibres.
Take smooth parameter strata and remove the images of any vertical
exceptional components of the simultaneous resolution. The relative
$\Q$-Cartier equality and the crepant coefficients less than one
then make the total-space pair klt as well. This verifies the
total-space hypothesis of \cite[Theorem~1.4]{CLZ26}.
After a generically finite base change and shrinking,
\cite[Theorems~1.4--1.6]{CLZ26} gives fibrewise small
$\Q$-factorializations and compatible numerical spaces, nef cones,
Mori chambers, and MMP steps. Its hypotheses hold: the fibres have
rational singularities and the stated vanishings, and every fibre
is of Fano type. Thus the very-general-fibre alternative in
\cite[Theorem~1.2]{CLZ26} applies.
For clarity, the finiteness-to-boundedness step is as follows. On
the geometric generic fibre of one stratum choose a small
$\Q$-factorialization. Finite generation supplies finitely many
small models, and finitely many faces and semiample contractions
on each of them. All this finite data spreads after a finite
extension and shrinking. The two inclusions needed for constancy
of a nef cone can be seen directly: spread its finitely many
semiample generators, which remain semiample on an open set, and
spread effective curves generating the dual polyhedral cone.
The divisor generators give one inclusion; the curve generators
give the reverse. The fibrewise $\Q$-factorializations and
compatible numerical spaces ensure that these comparisons take
place in the same space. The relative chamber/MMP compatibility
cited above then rules out further chambers on a fibre.
Here the source has first been made fibrewise $\Q$-factorial, so the
partial-MMP chamber description agrees with that from small modifications;
the converse direction in \cite[Theorem~1.6(3)]{CLZ26} applies. The
generic finite list of graphs consequently supplies the complete
list on an open set. Apply noetherian induction to the remaining
strata. This proves boundedness of the diagrams, not just of their
vertices.
We also check the last assertion. If $f:Y\dashrightarrow Z$ is a
non-extracting birational map, take a sufficiently ample Cartier
divisor $H$ on $Z$ and a general member avoiding the generic points
of the finitely many centers of divisors contracted by $f$.
Write $D$ for its strict transform on $Y$. The corresponding linear
system has no fixed prime component.
Moreover,
\[
R(Y,D)=R(Z,H).
\]
At a prime divisor retained by the map this is the same valuation
test on both sides. At a prime divisor contracted on $Y$, a
regular section of $\mathcal O_Z(mH)$ pulls back regularly, and
the chosen $H$ has order zero at that valuation. Normality then
gives equality of the section spaces. Hence $Z$ is the ample
model of a movable system on $Y$. A subsequent contraction is
handled by taking the pullback of an ample divisor from its base.
The Mori dream space chamber description therefore includes these
maps too.
If an intermediate target is not $\Q$-factorial, apply this description
to the composite from the initial $\Q$-factorial source; the diagram
also records the intermediate maps.
Finally take closures of the marked transforms on the finitely
many graph families, and resolve their union with the exceptional
loci. Proper images, inverse images, and irreducible components
of these finite-type families remain bounded after stratification.
This gives the stated marked and exceptional data.
\end{proof}
\begin{remark}[Bad loci in a bounded diagram]\label{bd:bad-loci}
For any of these finitely many contraction families, the proper
closed locus outside which a specified generic fibre property holds
can be chosen in the same bounded diagram. For example, use the
smooth locus in a conic or del Pezzo family. More generally, spread
a resolution of the geometric generic fibre and its very free
curves when that fibre is rationally connected. Smoothness and
deformation of the curves give a dense good open. Shrinking and
stratifying ensures that it is dense in each relevant fibre of the
parameter family. The complementary closed sets and their
preimages are then bounded. A prime divisor contained in such a
proper closed preimage is an irreducible component of that
preimage, so this observation bounds the divisors actually used
below; it makes no assertion about arbitrary subvarieties inside
a bounded variety.
\end{remark}
\subsection{Uniform geometric generic link diagrams}
\begin{theorem}[Uniform geometric generic link diagrams]
\label{bd:diagrams}
Bounded marked families of geometric generic diagrams can be chosen
with the following properties:
\begin{enumerate}[label=\textup{(\roman*)}]
\item There are such families depending only on the dimension so
that, for every specified birational map
$f:V/B\dashrightarrow V'/B'$ between terminal four-dimensional
Mori fibre spaces, there is an ordinary Sarkisov factorization of
$f$ whose individual geometric generic link diagrams belong to
these families.
\item For every fixed rational $0<c<1/4$, there are such families
depending only on the dimension and $c$ so that, for every smooth
projective threefold $W$ with a reduced SNC divisor $D$ and every
two given negative $(K_W+cD)$ programs with Mori fibre-space outputs,
there is a connecting log Sarkisov path whose individual geometric
generic link diagrams belong to these families.
\end{enumerate}
In \textup{(i)} the path induces the specified map $f$; in
\textup{(ii)} it induces the birational map determined by the two
given maps from $W$. The diagrams retain these identifications of
their function fields.
For a link with common base $Q$, put $F=\C(Q)$ and choose an
algebraic closure $\overline F$. The bounded diagrams are the
restrictions to the geometric generic fibre over $\overline F$.
They include the small models, both ends,
their base contractions, exceptional loci, marked transforms,
and bounded bad loci as in Remark~\ref{bd:bad-loci}.
In \textup{(i)} the marks are the exceptional and bad loci of the
link contractions; in \textup{(ii)} they also include the restrictions
of the transforms of the original divisor $D$.
Each induced end-base contraction $B\to Q$ is of Fano type
over $Q$. Its geometric generic fibre is integral and has
a rationally connected smooth proper model.
In \textup{(ii)} all marked prime divisors on a link model are
transforms of primes on the chosen common start: the maps
from that start do not extract. Higher starts with the mark
rule of Lemma~\ref{bd:higher-start} are allowed.
In both parts the families are independent of the start, its
resolution, the birational map,
the length of a path, or the K3 test collection.
\end{theorem}
\begin{proof}
Apply the geography above to a common smooth resolution of the
specified map in \textup{(i)}, and to the given common-start maps in
\textup{(ii)}, retaining their function-field identifications. Lemmas
\ref{bd:central} and \ref{bd:higher-start} give a central ample
model $Z^{\mathrm c}/Q$, reached by a small map from
$N^{\mathrm c}/Q$, with $\epsilon=1$ in the first case
and $\epsilon=1-c$ in the second. Put
$Z=Z^{\mathrm c}_{\overline F}$. Its dimension is at most four,
the negative log-canonical divisor is ample, and the same
discrepancy bound holds. Birkar's boundedness theorem
\cite[Theorem~1.1]{BirkarBAB21} applies: projective varieties
admitting an $\epsilon$-lc boundary with nef and big negative
log-canonical divisor form a bounded family. In particular
it bounds the underlying varieties $Z$ just constructed.
The marked boundary is bounded as well. In case \textup{(ii)},
write its restriction to $Z$ as $cD_Z$, and let $H$ be one of
the bounded very ample polarizations supplied by boundedness.
In fibre dimension $r$,
\[
cD_Z\cdot H^{r-1}\leq -K_Z\cdot H^{r-1}.
\]
The right side is bounded by the coefficient of the Hilbert
polynomial of the bounded polarized varieties. Thus $D_Z$
has bounded degree. Reduced cycles of bounded degree in a
fixed projective space form a bounded family, so the marked
central pairs are bounded. Marks vertical over $Q$ have empty
restriction to the generic fibre. Horizontal components may split
over $\overline F$, but their nonzero coefficients remain $c$
and their total degree is still bounded.
All the geometric generic link vertices and bases are
rational contractions of small $\Q$-factorializations of
these central pairs. Lemma~\ref{bd:contractions} supplies
bounded diagrams, and Remark~\ref{bd:bad-loci} supplies the
closed bad loci. Small loci do not acquire divisorial
components under the algebraic extension to the geometric
generic field. The same constructions apply over algebraically closed
fields of characteristic zero. For the family results stated over
$\C$, we use their finite-data spreading arguments and algebraically
closed base change to obtain the corresponding geometric generic
families here.
Every bound was taken from one of these fixed bounded
families, not by accumulating constants along a path.
For the assertion about base contractions, the central rank-two
model is of Fano type over $Q$ by Lemma~\ref{bd:central}.
This property survives small modifications and descends under
contractions, including fibre-type contractions
\cite[Lemma~2.8, with the relative convention after
Lemma--Definition~2.6]{ProkhorovShokurov09}.
It therefore holds for every end and its base $B\to Q$.
These are contractions of normal varieties, so their generic
fibres are geometrically integral in characteristic zero.
The rational connectedness theorem for klt log-Fano
contractions \cite{HM07Vertical}, applied also to a resolution,
gives the asserted rationally connected smooth proper models.
\end{proof}
\subsection{The low-discrepancy extractions}
The link diagrams have now been bounded. A different issue arises in
Proposition~\ref{sf:conic-end}: a conic base is controlled on its geometric
generic fibre, but an arbitrary smooth resolution need not be bounded
even there. The relative log MMP over that base will leave a geometric
generic model extracting only valuations of log discrepancy at most one,
as proved in that proposition. We prepare the bound for that surviving
model here; we do not bound the resolution itself.
The contraction lemma applies to maps which do not extract divisors.
We therefore first extract all exceptional prime valuations of log
discrepancy at most one. Any model extracting only some of them is a
birational contraction of this common model and falls under that lemma.
\begin{lemma}[Finite extraction list on an SNC pair]
\label{bd:snc-places}
Let $(R,\Gamma)$ be a smooth pair with SNC support and rational
coefficients $b_i<1$, allowing negative coefficients. There are
only finitely many exceptional divisorial valuations $F$ with
$a(F,R,\Gamma)\leq1$. They are obtained by a finite collection
of blow-ups of boundary strata. If the pairs and their SNC
components vary in a fixed log-smooth family with a fixed finite
coefficient set, all these valuations occur in finitely many such
relative constructions after stratification.
\end{lemma}
\begin{proof}
At the generic point of the center of $F$, of codimension $r$, use the regular parameters
and boundary coefficients in \eqref{bd:snc-discrepancy}.
If the center is not
a boundary stratum, at least one $b_i$ is zero; since an
exceptional center has $r\geq2$, the right side is strictly
greater than one. Thus a valuation in question centers at a
boundary stratum. Write its orders there as $w_i$. They satisfy
\[
\sum_i(1-b_i)w_i\leq1.
\]
There are finitely many possibilities because every $1-b_i$ is
positive.
Blow up the stratum. In a chart containing the center of the
valuation, choose an index with minimal order; the new orders
are that minimum and the differences from it. Zero differences
correspond to units unless the center lies in a further proper
subvariety of the exceptional stratum. The latter would be a
non-stratum center and is excluded by
\eqref{bd:snc-discrepancy} applied to the crepant SNC transform.
This transform is still klt, so its coefficients remain strictly less
than one and the same non-stratum exclusion applies at every step.
Thus, until the valuation itself becomes a divisor, its center
continues to be a stratum and the sum of its positive orders
strictly decreases. This is the usual monomial subtraction
algorithm, and proves both that the valuation is a stratum
valuation and that finitely many blow-up sequences suffice.
In a fixed family there are finitely many strata and bounded
orders; separate their geometric components and perform the
same finite sequences relatively.
\end{proof}
\begin{theorem}[Bounded terminalization diagrams]
\label{bd:terminalization}
Let $(T,\Lambda)$ range over a bounded collection of projective
klt log-Fano pairs with fixed finite rational coefficient set and
bounded marked support. There is a bounded collection of marked
diagrams containing a $\Q$-factorial crepant terminalization
extracting exactly the exceptional valuations
\[
a(F,T,\Lambda)\leq1.
\]
Every normal projective model which extracts only a subset of
these valuations, followed by non-extracting birational moves
and contraction morphisms, occurs among bounded marked
contraction diagrams of these terminalizations. The bound
depends on the given family, not on a resolution chosen to
construct one of the models.
\end{theorem}
\begin{proof}
Use Lemma~\ref{bd:cartier-strata} to obtain simultaneous crepant
log resolutions. Their coefficients are in a fixed finite set
strictly below one on each stratum. The exceptional divisors
already on the resolution are a finite list; all additional
valuations of log discrepancy at most one are in the uniformly
finite list of Lemma~\ref{bd:snc-places}. Extract all of them on
a common resolution.
The extraction theorem \cite[Corollary~1.4.3]{BCHM10} produces a
$\Q$-factorial model $\tau:T^{\mathrm t}\to T$ extracting exactly
this set. The extra restriction concerning non-klt centers in
that theorem is empty here. Its crepant boundary
$\Lambda^{\mathrm t}$ has coefficient $1-a(F,T,\Lambda)$ on an
extracted divisor, and hence is effective. By construction the
pair is terminal. Also
\[
-(K_{T^{\mathrm t}}+\Lambda^{\mathrm t})
=\tau^*\bigl(-(K_T+\Lambda)\bigr)
\]
is nef and big, so $T^{\mathrm t}$ is of Fano type. Apply this
construction on a geometric generic member and spread it; the
fixed extraction list, crepant equality, and the discrepancy
inequalities persist after shrinking. Thus the terminalizations
and all their marks form bounded families. This construction
does not resolve an unbounded set of arbitrarily chosen higher
models.
For a model $T'$ extracting a subset, every prime divisor on
$T'$ is already a valuation appearing on $T^{\mathrm t}$.
Consequently $T^{\mathrm t}\dashrightarrow T'$ is a birational
contraction. Lemma~\ref{bd:contractions}, applied to the bounded
Fano-type terminalization families, supplies its diagram and
every indicated subsequent contraction. In applying that lemma
one can either use the displayed weak log-Fano pair and a small
ample perturbation of its boundary on each stratum, or spread
a Fano-type boundary chosen on its generic member. Both give
families of klt log-Fano pairs on an open set. Include all the
extracted divisors as marks, even those with crepant coefficient
zero. The marked support on every output is then among the
bounded transforms already considered.
\end{proof}
\subsection{Choosing the ramification coefficient once}
\begin{lemma}[A uniform coefficient on bounded bases]
\label{bd:coefficient}
Let $(T,D)$ range over a bounded collection with $T$ a
projective klt Fano variety and $D$ a reduced divisor. Assume
$K_T$ and $D$ are $\Q$-Cartier and $D$ is numerically
proportional to $-K_T$. There is a rational $c_*>0$ such
that $(T,cD)$ is klt log Fano for every rational
$0<c<c_*$. This also holds when the numerical proportionality
is known over a characteristic-zero field before geometric
base change and both divisors are defined over that field.
\end{lemma}
\begin{proof}
Zero-dimensional members have $D=0$ and impose no restriction on $c$;
we treat the positive-dimensional members below.
Apply Lemma~\ref{bd:cartier-strata} to the log-Fano pairs
$(T,0)$ with additional marked divisor $D$.
On finitely many simultaneous log resolutions the crepant
coefficients for $(T,0)$ and the coefficients of $\pi^*D$
are fixed. Write $a_E=a(E,T,0)>0$ and $m_E$ for the
coefficient of $\pi^*D$ on an exceptional component.
Choosing $c<1/2$ and $cm_E<a_E/2$ for every $m_E>0$
makes every coefficient of the crepant pullback of $cD$
strictly less than one. These finitely many inequalities
give a uniform positive klt bound for $c$.
For ampleness use a bounded very ample divisor $H$ and
let $r=\dim T$. A common multiple $mK_T$ is Cartier
on each of the finitely many strata. Thus
\[
k=-K_T\cdot H^{r-1}>0,\qquad mk\in\Z_{>0}.
\]
The degree $e=D\cdot H^{r-1}$ is a bounded nonnegative
integer. Numerical proportionality says
$D\equiv(e/k)(-K_T)$. Since $e/k\leq me$, choosing
$cme<1/2$ for all these data gives
\[
-(K_T+cD)\equiv (1-ce/k)(-K_T),
\]
which is ample. Taking the minimum of the finitely many
rational choices gives $c_*$.
Finally numerical equivalence between divisors defined
over the original field persists geometrically. Any
geometric curve is defined over a finite extension; all
its conjugates have the same intersection with such a
divisor. Its orbit descends to a curve cycle over the
original field, so a zero intersection downstairs
forces zero on every conjugate. The preceding argument
therefore applies after the geometric base change.
\end{proof}
For the next corollary, let $V\to T$ be a conic Mori fibre space and
let $\alpha\in\Br(\C(T))[2]$ be the class of its generic conic.
At a prime divisor $P\subset T$, its Brauer residue belongs to
$H^1(\C(P),\Z/2)$. Define $D_T$ to be the reduced sum of the primes
with nonzero residue; this is the \emph{full visible divisorial
ramification support} on this model. For a map $T\to Q$, put
$F=\C(Q)$. On $T_{\overline F}$ we retain the divisor
$(D_T)_{\overline F}$ obtained by restricting and extending this
original support. A residue can become zero after extension of
constants; its original supporting divisor remains marked.
\begin{corollary}[Order of the uniform choices]
\label{bd:choice-order}
For the conic-base ends $V/T$ of ordinary fourfold links in the
paths of Theorem~\ref{bd:diagrams}, over $Q$ with $\dim Q\leq1$,
a single rational number
\[
0<c<1/4
\]
can be fixed such that the geometric generic pair
$(T_{\overline F},c(D_T)_{\overline F})$, with $F=\C(Q)$, is klt
log Fano. Here $D_T$ is the full visible divisorial ramification
support defined on the original conic base above.
Fix this number before applying the log part of
Theorem~\ref{bd:diagrams} or
Theorem~\ref{bd:terminalization}. All resulting
large-degree exclusions have a common threshold
independent of the K3 test subcollection and of the chosen
Sarkisov path.
\end{corollary}
\begin{proof}
First use only the ordinary part of
Theorem~\ref{bd:diagrams}. At the indicated end the tower is
$N_i\xrightarrow{e_i}V\to T\to Q$. Since $\dim T=3$ and
$\dim Q\leq1$, the last contraction has positive relative
dimension and $\rho(T/Q)\geq1$. Equation~\eqref{bd:rank-tower} gives
\[
2=\rho(N_i/V)+1+\rho(T/Q).
\]
Consequently $\rho(N_i/V)=0$ and $\rho(T/Q)=1$: the arrow $e_i$
is the identity and $T\to Q$ is an elementary contraction of fibre
type. This calculation takes place over $Q$ before geometric
generic base change.
The geometric generic model $T_{\overline F}$ and its conic
projection occur in the bounded diagram. The restriction of the
original visible ramification is contained in the
bounded bad locus of this projection, and its divisorial
support is a union of the finitely many divisorial
components there. These marked supports are therefore
bounded.
Apply \cite[Lemma~4.6]{Druel16} to the elementary conic Mori
contraction $V\to T$. Its source is terminal and $\Q$-factorial,
so its base $T$ is $\Q$-factorial.
Divisors on $T_F$ extend by closure to Weil divisors on $T$,
which are $\Q$-Cartier. Restriction therefore surjects onto the
generic numerical divisor space. The relative rank-one condition
and the nonzero restriction of an ample class give $\rho(T_F/F)=1$.
Fano type descends under contractions
\cite[Lemma~2.8]{ProkhorovShokurov09}; hence $T_F$ and
$T_{\overline F}$ are of Fano type. Over the original generic field
$F$, rank one implies that $-K_{T_F}$ is ample: it is the sum of
an ample log-anticanonical class and an effective class.
It also implies numerical proportionality of $(D_T)_F$ and
$-K_{T_F}$. These two divisors are $\Q$-Cartier before, and
hence after, base change. Their numerical proportionality persists
over $\overline F$ by Lemma~\ref{bd:coefficient}, even if the
geometric Picard rank increases. That lemma
provides a uniform $c_*$ from these ordinary diagrams.
Choose a rational $c<\min\{1/4,c_*\}$ once. With this
fixed value the marked log links have the uniform bound
of Theorem~\ref{bd:diagrams}, and the bounded
$(T_{\overline F},c(D_T)_{\overline F})$ data have the extraction bound of
Theorem~\ref{bd:terminalization}. Only finitely
many bounded families, dimensions, and bounded-degree
cover constructions are used subsequently: the exceptional and
bad-locus components in Section~\ref{sl:section}, and the
ramification and del Pezzo class covers in Section~\ref{sf:section}.
Their branch and degree bounds are checked there. Take the
maximum of their thresholds in
Theorem~\ref{bd:exclusions}. Every bound depends
only on those families and this chosen $c$, proving
the asserted independence.
\end{proof}
==== END Irrational-cubic-fourfolds-with-geometric-K3-categories//bounds.tex ====
==== BEGIN Irrational-cubic-fourfolds-with-geometric-K3-categories//cubics.tex ====
\section{Cubic fourfolds of admissible discriminant}
\label{cub:section}
We now combine the two birational calculations. The first records an
integral transcendental contribution in a smooth factorization; the second
rules out that contribution along a Mori factorization, uniformly as the
degree tends to infinity.
On a cubic $X$, write $H$ for the hyperplane divisor and
$h=c_1(\mathcal O_X(1))$ for its cohomology class.
\begin{proposition}[An irrationality criterion]
\label{cub:criterion}
There is an integer $d_0$ with the following property. Let $X$ be a smooth
complex cubic fourfold, and let
\[
A_X=H^4(X,\Z)\cap H^{2,2}(X),
\qquad T_X=A_X^\perp\subset H^4(X,\Z).
\]
Suppose that
\[
\operatorname{rank}T_X=21,\qquad
d=|\det T_X|>d_0,\qquad
\operatorname{End}_{\mathrm{Hdg}}(T_X\otimes\Q)=\Q.
\]
Then $X$ is not rational.
\end{proposition}
\begin{proof}
Choose $d_0>2$ at least as large as the uniform bound in
Theorem~\ref{sf:vanishing}. Its uniformity allows the collection of K3
surfaces to be restricted to any specified integral transcendental
Hodge-isometry class.
Suppose that $f\colon\mathbb P^4\dashrightarrow X$ is birational. Take as
test collection the Picard-number-one K3 surfaces of degree $d$ whose
transcendental Hodge lattice, after the shift of Hodge types and reversal
of the cup pairing, is isometric to $T_X$. These surfaces have no
nonidentity automorphisms by Lemma~\ref{gw:k3-center}.
The assumed rationality makes $X$ rationally connected, and the cubic
Hodge calculation gives $h^{3,1}(X)=1$
\cite[Section~2.1]{Hassett00}. Thus all hypotheses of
Proposition~\ref{gw:net-center} hold, and it gives
\begin{equation}
\label{cub:one}
c_u(f)=1.
\end{equation}
Both varieties are Mori fibre spaces over a point. Indeed, they are
smooth, and integral weak Lefschetz and the exponential sequence give
$\operatorname{Pic}(X)=\Z\mathcal O_X(1)$, while adjunction gives
$-K_X=3H$ \cite[Chapter~1, Section~1.1, Lemma~1.6 and
Corollary~1.9]{HuybrechtsCubic2023}. The ordinary Sarkisov program therefore factors $f$ into the
links used in Theorem~\ref{sf:vanishing}. Applying that theorem to this
factorization yields $c_u(f)=0$, contradicting \eqref{cub:one}.
\end{proof}
\subsection{The classical and categorical inputs}
We use the following established results in their untwisted form, with
the labellings and admissible discriminants defined in the introduction.
Hassett's period and lattice theorems
\cite[Theorems~1.0.1, 1.0.2 and 5.1.3; Corollary~5.2.4]{Hassett00}
give the nonempty irreducible divisor $\mathcal C_d$ when
$d>6$ and $d\equiv0,2\pmod6$. For admissible $d$, a labelled cubic has
an associated primitively polarized K3 surface $(S,L)$ of degree $d$,
with a Hodge isometry
\begin{equation}
\label{cub:associated}
K^\perp \simeq L^\perp(-1),
\end{equation}
where the surface cup pairing is reversed. Hassett's marked space also
records an identification of the distinguished rank-two lattice $K$
with a fixed abstract lattice, preserving $h^2$. The map forgetting
this identification is finite \cite[Section~5.2 and
Proposition~5.2.1]{Hassett00}. His marked space admits an open immersion
into the moduli space of degree-$d$ polarized K3 surfaces
\cite[Corollary~5.2.4]{Hassett00}.
For such a cubic, there exists a smooth projective K3 surface $S'$ and
an exact $\C$-linear equivalence
\begin{equation}
\label{cub:categorical}
\operatorname{Ku}(X)\simeq D^b(\operatorname{Coh}(S')).
\end{equation}
This is \cite[Corollary~29.7]{BLMPNS21}; it applies to every cubic having
a Hodge-theoretically associated K3 surface. Its target is the ordinary
derived category. Already
\cite[Theorem~1.1]{AddingtonThomas14} gives \eqref{cub:categorical} on a
Zariski open dense subset of each admissible $\mathcal C_d$, which would
suffice below. Those equivalences are Fourier--Mukai equivalences, so
they have the exactness and $\C$-linearity in Theorem~\ref{thm:main}.
\subsection{The very general transcendental lattice}
\begin{lemma}
\label{cub:generic}
For each admissible $d$, a very general point $X\in\mathcal C_d$
satisfies
\begin{equation}
\label{cub:generic-data}
\operatorname{rank}A_X=2,\quad
\operatorname{rank}T_X=21,\quad
|\det T_X|=d,\quad
\operatorname{End}_{\mathrm{Hdg}}(T_X\otimes\Q)=\Q.
\end{equation}
Moreover, $T_X\otimes\Q$ is simple. All these assertions hold
simultaneously with \eqref{cub:categorical} outside a countable union
of proper closed algebraic subsets of $\mathcal C_d$.
\end{lemma}
\begin{proof}
We spell out the exceptional sets, including their algebraicity.
Pass to Hassett's marked space and then add finite level structure.
Fix a smooth irreducible component that is finite and surjective over
a dense open $U\subset\mathcal C_d$, and denote it by $U'$.
The rank-two marking makes $K$ constant. Its orthogonal
$\mathbb T=K^\perp$ is an integral polarized variation on $U'$;
we trivialize this complementary local system only on period charts.
All exceptional loci below are constructed on this one fixed cover.
The cubic Hodge numbers and middle lattice are
\[
h^{3,1}=h^{1,3}=1,\quad h^{2,2}=21,\quad b_4=23,
\qquad
H^4(X,\Z)\simeq \langle1\rangle^{\oplus21}
\oplus\langle-1\rangle^{\oplus2};
\]
see \cite[Section~2.1 and Proposition~2.1.2]{Hassett00}.
Thus $\mathbb T$ has rank $21$ and signature $(19,2)$ for the cubic
cup pairing. In a marked period chart its period line varies in an
open subset of
\[
\mathcal D_T=
\left\{[\omega]\in\mathbb P(T\otimes\C):
(\omega,\omega)=0,\ (\omega,\overline\omega)<0\right\}.
\]
Here $T$ denotes the underlying fixed lattice in that chart. The
openness follows equivalently from the associated K3 period description
\eqref{cub:associated}. The domain $\mathcal D_T$ is analytically open in a smooth
irreducible projective quadric, and this quadric spans
$\mathbb P(T\otimes\C)$.
For $0\ne t\in T\otimes\Q$, consider the locus in $U'$ where some
flat translate of $t$ is a Hodge class. After clearing denominators
and Tate twisting, \cite[Theorem~1.1 and Corollary~1.3]{CDK95} makes
this locus closed algebraic, with finitely many local branches.
Here one fixed denominator works for the whole monodromy orbit, and
the flat polarization has the same value on all its translates.
The cited corollary obtains this orbit locus from images of connected
components of the bounded-norm space of Hodge classes, which the
theorem makes finite over the base.
In a period chart each branch lies in
$(t')^\perp\cap\mathcal D_T$ for a nonzero flat translate $t'$.
Each such section is proper because the period domain spans
$T\otimes\C$; hence the global locus is proper as well.
There are countably many rational $t$. Since
$H^4(X,\Q)=K_\Q\oplus K_\Q^\perp$, excluding these loci removes every
Hodge class outside $K_\Q$. Thus $A_X\otimes\Q=K\otimes\Q$;
saturation gives $A_X=K$ and consequently $\operatorname{rank}T_X=21$.
The middle lattice is unimodular, and both $A_X$ and $T_X$ are
primitive. To recall the determinant argument, write $L=H^4(X,\Z)$.
Every homomorphism $A_X\to\Z$ extends to $L$, since $L/A_X$ is
torsion-free. Unimodularity therefore makes the restriction map
$L\to A_X^*$ surjective with kernel $T_X$. Reducing modulo $A_X$
gives
\[
L/(A_X\oplus T_X)\simeq A_X^*/A_X.
\]
Interchanging $A_X$ and $T_X$ gives the same quotient as $T_X^*/T_X$.
The two discriminant groups have the same order; hence
$|\det T_X|=\det A_X=d$.
Next fix a nonscalar $a\in\operatorname{End}_{\Q}(T\otimes\Q)$.
If $a$ is a Hodge endomorphism, then $\C\omega$ is an eigenline for
$a$. Hence its Hodge locus is contained in
\[
\bigcup_{\lambda}\mathbb P\ker(a-\lambda)\cap\mathcal D_T.
\]
This is a finite union of proper linear sections of the period quadric.
Indeed, an eigenspace equal to $T\otimes\C$ would make $a$ scalar;
and an irreducible quadric spanning the ambient vector space cannot
be contained in a finite union of proper linear subspaces.
There are countably many rational endomorphisms $a$. Apply the same
argument to the weight-zero polarized variation
$\operatorname{End}(\mathbb T)$, clearing denominators. The locus where
some flat translate of $a$ is Hodge is closed algebraic with finitely
many local branches. Monodromy acts by conjugation and preserves
nonscalarity, so the eigenline argument makes every local branch
proper. These global loci are therefore proper. Excluding them gives
$\operatorname{End}_{\mathrm{Hdg}}(T_X\otimes\Q)=\Q$.
For completeness, simplicity also follows directly from the Hodge
numbers. By polarizability, a rational Hodge substructure has a Hodge
complement. In any nontrivial direct-sum decomposition of $T_X\otimes\Q$,
only one summand could contain the one-dimensional $(3,1)$ part, and
that summand would also contain its conjugate. Every other summand
would consist entirely of rational $(2,2)$ classes, contrary to the
definition of $T_X$.
Finally take the finite images in $U$ of the exceptional loci in $U'$.
These images are closed and proper, since $U'$ is irreducible and
finite over $U$. Their closures in $\mathcal C_d$ are still proper.
Adding $\mathcal C_d\setminus U$ gives a countable union of proper
closed algebraic subsets, whose complement is nonempty over $\C$.
The categorical
assertion holds throughout $\mathcal C_d$ by
\eqref{cub:categorical}; if one uses only the Addington--Thomas theorem,
one additionally removes the complement of its dense open subset.
This proves the simultaneous assertion.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:main}]
Let $d_0$ be as in Proposition~\ref{cub:criterion}, and fix any admissible
$d>d_0$. The divisor $\mathcal C_d$ is nonempty. By
Lemma~\ref{cub:generic}, outside a countable union of proper closed
algebraic subsets of this divisor, the lattice $T_X$ has rank $21$,
absolute determinant $d$, and rational Hodge endomorphism ring $\Q$.
The same lemma supplies an exact $\C$-linear equivalence
\[
\operatorname{Ku}(X)\simeq D^b(\operatorname{Coh}(S))
\]
for a smooth projective K3 surface $S$. Proposition~\ref{cub:criterion}
applies to each such $X$ and proves that it is irrational.
The integer $d_0$ was fixed before choosing $d$, so the conclusion holds
very generally in every admissible $\mathcal C_d$ with $d>d_0$. Its
ineffectiveness is inherited from the uniform bound in
Theorem~\ref{sf:vanishing}. Such a cubic satisfies the categorical
condition of Kuznetsov's conjecture and fails its rationality condition;
hence the categorical-to-rational implication is false.
\end{proof}
The sequence $d=2\cdot7^j$ gives an explicit infinite subfamily of the
admissible discriminants. For $j\geq1$, one has $d>6$ and
$d\equiv2\pmod6$; the prime $2$ occurs with exponent one, $3$ does not
divide $d$, and the only odd prime divisor is $7\equiv1\pmod3$.
Thus every member satisfies \eqref{cub:admissible}, and the theorem
applies to all sufficiently large members of this sequence.
\subsection{Proofs of the consequences}\label{cub:consequences}
We finish by combining Theorem~\ref{thm:main} with the established K3
and zero-cycle results used in the introduction.
\begin{proof}[Proof of Corollary~\ref{cor:associated-k3}]
For admissible \(d\), the classical input \eqref{cub:associated} supplies
the indicated polarized K3 surface for every labelled cubic in
\(\mathcal C_d\). Theorem~\ref{thm:main} supplies irrationality on the
stated very-general set for \(d>d_0\). Since \(K\) contains \(h^2\), the
isometry gives
\[
L^\perp(-1)\simeq K^\perp\hookrightarrow
(h^2)^\perp=H^4(X,\Z)_{\mathrm{pr}}.
\]
The inclusion is primitive: if a nonzero integer multiple of an integral
class in \((h^2)^\perp\) is orthogonal to \(K\), then the class itself is
orthogonal to \(K\). This is the primitive polarized K3 Hodge embedding
in the cited formulation, with the stated Tate twist and pairing reversal.
\end{proof}
\begin{proof}[Proof of Corollary~\ref{cor:hilbert-square}]
For \(n\ge2\), the integer \(d=2n^2+2n+2\) is greater than six and is
congruent to zero or two modulo six. Taking \(a=1\) realizes Addington's
condition \((***)\), namely
\(d=(2n^2+2n+2)/a^2\) for integers \(n,a\) with \(a\ne0\).
That condition implies his condition \((**)\), the remaining divisibility
restrictions in \eqref{cub:admissible}; hence \(d\) is admissible.
Addington's Theorem~2 then supplies the stated birationality for every
\(X\in\mathcal C_d\), with a projective K3 surface by his convention
\cite[condition~\((***)\), Theorem~2, and the convention in the introduction]{Addington16}.
Theorem~\ref{thm:main} supplies irrationality for a very general such
\(X\) when \(d>d_0\). Finally, the values \(2n^2+2n+2\) are strictly
increasing and unbounded for \(n\ge2\), so infinitely many exceed \(d_0\).
\end{proof}
\begin{proof}[Proof of Corollary~\ref{cor:chow-diagonal}]
First let \(X\in\mathcal C_d\) be any cubic with a labelling \(K\) of
admissible discriminant \(d\). The intersection form satisfies
\((h^2,h^2)=\int_Xh^4=3\), so \(h^2\) is primitive in the integral middle
lattice: if \(h^2=m\eta\) for an integral class \(\eta\), then \(m^2\)
divides \(3\). It is therefore primitive in \(K\) and extends to a basis
\((h^2,\sigma)\) of this rank-two lattice.
Voisin's Theorem~5.6 is stated for the lattice \(P\) generated by the
independent integral Hodge classes \(h^2\) and \(\sigma\), which are
algebraic on a smooth cubic fourfold as recalled in its proof
\cite[Theorem~5.6 and its proof]{Voisin17}. Here \(P=K\), and its
discriminant in that theorem is exactly
\[
D(\sigma)=
\det\begin{pmatrix}
3 & h^2\mathbin{\cdot}\sigma\\
h^2\mathbin{\cdot}\sigma & \sigma^2
\end{pmatrix}
=\det K=d.
\]
Admissibility includes \(4\nmid d\). Voisin's theorem therefore gives
universal triviality of \(\operatorname{CH}_0(X)\) for every such labelled
cubic, without requiring \(K\) to be the entire lattice of Hodge classes
or \(X\) to be very general. The field-extension scope and the equivalence
with the displayed integral Chow-theoretic decomposition are given in the
introduction and equation~(1) of the same paper \cite{Voisin17}.
Theorem~\ref{thm:main} supplies irrationality for the stated very general
cubics when \(d>d_0\).
\end{proof}
==== END Irrational-cubic-fourfolds-with-geometric-K3-categories//cubics.tex ====
==== BEGIN Irrational-cubic-fourfolds-with-geometric-K3-categories//gw.tex ====
\section{Whole transcendental lattices under factorization}\label{gw:section}
We construct an operator algebra which distinguishes the whole transcendental
contribution of each surface center. Its role is to make the classical
integral blowup decomposition compatible with cancellation in an arbitrary
weak factorization.
Quantum blowup decompositions also appear in Iritani's decomposition of
quantum $D$-modules and quantum cohomology $F$-manifolds
\cite[Theorem~1.1 and Corollary~1.2]{Iritani25}.
That decomposition uses a formal Novikov extension and a change of
variables. Here we study the rational operator algebra in the classical
blowup splitting and its saturated integral transcendental summands.
\subsection{Hodge structures and the operator algebra}
For a smooth projective variety $Y$, write
\[
H^{2i}(Y,\Q)_{\mathrm{Hdg}}
=H^{2i}(Y,\Q)\cap H^{i,i}(Y).
\]
If $M$ is a rationally connected smooth projective fourfold, set
\[
L_M=H^4(M,\Z)/\mathrm{tors},\qquad
T_M=L_M\cap H^4(M,\Q)_{\mathrm{Hdg}}^{\perp}.
\]
The orthogonal is taken for the cup form $q_M$. For a smooth projective
surface $Z$, use the analogous definition
\[
T_Z=(H^2(Z,\Z)/\mathrm{tors})\cap
H^2(Z,\Q)_{\mathrm{Hdg}}^{\perp},
\]
with surface cup form $q_Z$. We always divide out torsion before discussing
integral lattices. When comparing with degree four, $T_Z(-1)$ means the Tate
shift to weight four, equipped with the form $-q_Z$.
\begin{lemma}\label{gw:hodge}
The forms on $T_M$ and $T_Z$ are nondegenerate. The rational Hodge structure
$T_{M,\Q}$ is generated by its $(3,1)$ and $(1,3)$ subspaces, and
$T_{Z,\Q}$ is generated by its $(2,0)$ and $(0,2)$ subspaces. The only
off-diagonal types in the even cohomology of $M$ are $(3,1)$ and $(1,3)$ in
degree four. If $i:Z\hookrightarrow M$ is a surface embedding, then
\[
i^*T_{M,\Q}=0,\qquad i_*T_{Z,\Q}=0.
\]
\end{lemma}
\begin{proof}
Rational connectedness gives $H^{p,0}(M)=0$ for $p>0$
\cite[Theorem~30]{AraujoKollar02}.
We use the Hodge decomposition, functoriality and Lefschetz polarizations
in \cite[Sections~2.1--2.2]{VoisinHodge2016}; the Hodge-complement
statement is Lemma~1 there.
Thus $H^2(M,\Q)$ is of type $(1,1)$; hard Lefschetz gives the analogous
statement in degree six. The remaining assertion about types follows from
Hodge symmetry. In degree four all nonprimitive Lefschetz summands are
rational Hodge classes. Hodge--Riemann therefore makes their orthogonal
primitive transcendental part nondegenerate. The same argument in degree
two applies to a surface, using an ample class to split off its
nonprimitive part. Alternatively, the nondegeneracy on the Hodge-class
space follows on each primitive Lefschetz summand from definiteness.
By semisimplicity of polarizable rational Hodge structures, the substructure
generated by the stated off-diagonal types has a Hodge complement in the
transcendental structure. That complement would have only diagonal type,
and hence would consist of rational Hodge classes. It is zero by the
definition and nondegeneracy of the transcendental part. Finally, $i^*$
takes the off-diagonal part of $H^4(M)$ into $H^4(Z)$, which has only type
$(2,2)$; and $i_*$ takes the off-diagonal part of $H^2(Z)$ into $H^6(M)$,
which has only type $(3,3)$. Both maps vanish on the generating types and
therefore on the whole corresponding rational Hodge structure.
\end{proof}
All Gromov--Witten invariants below are connected, of genus zero, and use
cohomology with rational coefficients. The \emph{Hodge difference} of a
class of type $(p,q)$ is $p-q$. All degrees are collected by numerical
curve class: an invariant denotes the sum over homology classes with the specified numerical class. This
sum is finite in every bounded ample degree, and
all degree comparisons below use an integral ample class. Other than the
two specified variable insertions, every insertion is a rational class of
diagonal Hodge type. Descendant powers at all ordinary markings, including
the two variable markings, are arbitrary nonnegative integers.
These two markings carry distinct formal labels in every weighted list
and degeneration graph. The labels remain distinct even when the two
vectors agree or one is zero. Automorphism and gluing coefficients thus
depend on the fixed discrete data, independently of the variable vectors.
\begin{definition}\label{gw:algebra}
For every invariant
\[
B(u,v)=
\left\langle\tau_a(u),\tau_b(v),
\tau_{a_1}(\gamma_1),\ldots,\tau_{a_n}(\gamma_n)
\right\rangle^M_{0,\beta},\qquad u,v\in T_{M,\Q},
\]
let $A_B\in\operatorname{End}_{\Q}(T_{M,\Q})$ be determined by
$B(u,v)=q_M(A_Bu,v)$. Let $G_M$ be the unital rational algebra generated by
all such operators. On the zero space we take the zero algebra.
\end{definition}
The virtual classes, evaluation maps, and cotangent classes defining these
invariants respect Hodge structures. Thus $A_B$ is a Hodge endomorphism.
Interchanging the two variable markings supplies its adjoint. In particular,
every expression in $G_M$ is a rational Hodge endomorphism. By
Lemma~\ref{gw:hodge}, equality of two such expressions can be checked on
$(3,1)$, by pairing their images with $(1,3)$. We will use the analogous
test on surfaces, after the indicated Tate shift.
Accordingly, when analyzing a tensor on copies of $T_M$ or $T_Z$, we first
test its two inputs on opposite off-diagonal types and then use this
Hodge-generation statement to recover the full rational tensor.
\begin{theorem}[Operator and integral blowup decomposition]\label{gw:blowup}
Let $\pi:Y\to M$ be the blowup of a connected smooth center $Z$ of
codimension $r\geq2$ in a rationally connected smooth projective fourfold.
If $Z$ is a surface, the classical embeddings identify
\begin{equation}\label{gw:split-equation}
(T_Y,q_Y)=(T_M,q_M)\perp(T_Z(-1),-q_Z)
\quad\text{over }\Z,
\qquad
G_Y=G_M\oplus\Q\,\operatorname{id}_{T_Z}.
\end{equation}
The second factor is omitted when $T_Z=0$. For a point or curve center,
$T_Y=T_M$ integrally and $G_Y=G_M$.
\end{theorem}
Here and subsequently a product decomposition of operator algebras means
that the two factors act independently on the displayed summands, with
zero mixed blocks. In particular, the assertion is stronger than the
reconstruction of individual numerical invariants.
Retain the blowup $\pi:Y\to M$ of Theorem~\ref{gw:blowup} throughout
its proof. Write $i:Z\hookrightarrow M$ and $j:D\hookrightarrow Y$
for the center and exceptional-divisor embeddings, where
$D=\mathbb P_Z(N_{Z/M})$. Let $p:D\to Z$ and
$\xi=c_1(\mathcal O_D(1))$. Put $W=T_{Z,\Q}$ if $Z$ is a surface,
and $W=0$ otherwise.
We first identify the integral summands in the theorem. The classical
blowup formula has explicit Hodge-compatible integral maps
\cite[Section~2.1, Lemmas~9--10]{Schreieder14}. In the surface case it reads
\[
H^4(Y,\Z)/\mathrm{tors}
=\pi^*(H^4(M,\Z)/\mathrm{tors})
\oplus j_*p^*(H^2(Z,\Z)/\mathrm{tors}).
\]
The summands are orthogonal, since $p_*1=0$, and on the exceptional summand
$j^*j_*=-\xi$ and $p_*\xi=1$ give $-q_Z$. After rationalization this is a
direct sum of Hodge structures, so its Hodge-class subspace splits as well.
Taking the orthogonal and intersecting with the displayed integral direct
sum gives the integral transcendental decomposition. For a curve or
point center the additional degree-four cohomology is diagonal and
contributes no transcendental summand.
The remaining assertion concerns the operator algebra on this fixed
splitting. We first prove
$G_Y\subseteq G_M\oplus\Q\operatorname{id}_W$: there are no mixed
operators, the exceptional block is scalar, and the old block lies in
$G_M$. An exceptional-line invariant then supplies the projector onto
$W$, separating that factor from its complement. Finally we recover
every operator in $G_M$ on the old summand. The two inclusions require
different relative reconstructions: the first uses ordinary insertions
pulled back from $M$, whereas recovery allows arbitrary ordinary
insertions on $Y$. When $W=0$, only the two inclusions are needed.
The next three subsections carry out these steps; the final subsection
uses the product decomposition to cancel whole integral lattices along
a weak factorization.
\subsection{Classical degeneration and local path tensors}
Both reconstruction arguments express a two-variable tensor as contractions
of connected relative tensors. The genus-zero tree reduces these
contractions to the path joining the two variable markings. Along that
path, local projective-bundle vertices supply scalar pairings between
copies of $W$. We first specify the connected-factor and descendant
conventions needed for these statements.
We use expanded relative stable maps in the sense of Jun Li
\cite{Li01,Li02}. An ordinary marking carries an evaluation class on the
target; a relative marking carries a positive contact order and an
evaluation class on the boundary divisor. In a degeneration graph the
relative markings are the joining roots. A descendant at an ordinary marking is the first Chern
class of the cotangent line of the actual source curve there. These are the
same cotangent lines in every relative theory used below. Gluing relative
markings to nodes changes no neighborhood of an ordinary marking; hence
these cotangent lines restrict to the corresponding lines on the relative
factors. On a smooth fibre they are the usual absolute descendant lines.
We never require a descendant at a joining relative marking.
We require the degeneration formula with a product of \emph{connected}
relative factors. A disconnected source can share one expanded target,
so this product requires a virtual-class theorem. Here is the precise
chain of results we use. Apply the ordinary-descendant degeneration formula
of Abramovich--Fantechi \cite[Theorem~0.1 and Section~5]{AF16}.
For fixed labelled connected-component data
$\Xi=\coprod_\nu\Gamma_\nu$, write $\mathcal K_{\Gamma_\nu}(X,D)$ for
the relative-map space with that connected degree, genus, and marking data,
and $\mathcal K_\Xi(X,D)$ for the space with all those components sharing
an expanded target. Their contraction morphism
\[
\epsilon:\mathcal K_\Xi(X,D)\longrightarrow
\prod_\nu\mathcal K_{\Gamma_\nu}(X,D)
\]
separates and stabilizes the components, and satisfies
\[
\epsilon_*[\mathcal K_\Xi(X,D)]^{\mathrm{vir}}
=\boxtimes_\nu[\mathcal K_{\Gamma_\nu}(X,D)]^{\mathrm{vir}}
\]
by \cite[Proposition~4.14]{AF16}. Corollary~4.15 there gives the
ordinary-descendant product formula, retaining all relative evaluation
weights. This corollary supplies descendant compatibility under the
separating and stabilizing map $\epsilon$.
Its hypothesis that each connected component has a marking
holds here: each vertex of a nontrivial connected degeneration graph has
a joining root. A one-vertex term needs no product decomposition.
Next apply \cite[Theorem~1.1.2]{AMW14} to each connected factor to identify
it with the Jun Li relative invariant.
These steps preserve our cotangent convention. The ordinary lines in
\cite[Definition~4.1 and Section~4.3]{AF16} agree with those on the actual
coarse source near ordinary markings. The comparison morphism from
Abramovich--Fantechi to Li contracts no source components
\cite[Section~4.1]{AMW14}; its virtual pushforward and the projection
formula therefore identify the actual-source descendants as well as the
evaluation classes. We need no descendant at a joining root and no
comparison for disconnected Li moduli. This also explains why we use the
product theorem for maps to a fixed relative target; no product rule for
rubber targets is asserted.
The resulting formula expresses an absolute invariant as a finite sum
over matching connected relative data. Each edge contracts a pair of
relative weights with the diagonal of the double divisor. The coefficient
is the product of contact orders divided by the automorphism and labeling
factors. A connected genus-zero source gives a tree of connected vertices.
We retain the two variable insertions throughout, so the formula is an
identity of bilinear tensors.
The two normal-cone degenerations used in the reconstruction have
auxiliary components
\[
P=\mathbb P_Z(N_{Z/M}\oplus\mathcal O_Z),\qquad
P_D=\mathbb P_D(N_{D/Y}\oplus\mathcal O_D).
\]
Degenerating $M$ along $Z$ gives the pair $(Y,D)$ joined to $P$ along its
infinity divisor $D$; its zero section is $Z$.
Degenerating $Y$ along $D$ gives $(Y,D)$ joined to $P_D$ along its infinity
section, another copy of $D$; its zero section is also a copy of $D$.
Both auxiliary components, and their expansions, project to $Z$.
All classes of Hodge difference $\pm2$ on $D$, on
$P$, and on the projective bundles
appearing when one degenerates along $D$, come from copies of $W$ multiplied
by tautological classes. If $W=0$ there are no such classes.
Multiplication of one of these copies of $W$ by a fixed diagonal class
preserves the sum of the copies, and is scalar between its degree-compatible
copies. Indeed, the bundle formula reduces the assertion to base
multiplication on $Z$; a positive-degree diagonal base class annihilates
$W$, as is seen on $(2,0)$ and $(0,2)$ and then by
Lemma~\ref{gw:hodge}. A product of two $W$ variables is $q_Z(u,v)$ times
the point class, with any tautological factors retained. We will use these
facts when tail outputs combine cohomology labels.
\begin{lemma}[Local tensors]\label{gw:local}
On any of the projective-bundle pieces just described, a connected
genus-zero invariant, either absolute or relative to the indicated infinity
boundary, with two variable insertions in the corresponding
copies of $W$ and all other insertions rational and diagonal is a
rational scalar multiple of $q_Z$ on the corresponding copies of $W$.
It vanishes if the projected curve class on $Z$ is nonzero. These statements
allow ordinary descendants and expanded targets.
\end{lemma}
\begin{proof}
There is nothing to prove if $W=0$. Otherwise $p_g(Z)>0$, so $Z$ is not
uniruled. A fixed-degree connected genus-zero stable map with positive
projected degree has projected image a connected union of rational curves.
For each irreducible component of its finite-type moduli space the evaluation
image in $Z$ has dimension at most one: a dominating evaluation image would
give a covering family of rational curves on $Z$. There are finitely many
such components for the fixed discrete data. A holomorphic two-form pulls
back to zero through each of their evaluation maps, since their images lie
in curves or points. The bilinear tensor, with all other data fixed, is
rational and respects Hodge structures. Its associated Hodge endomorphism
therefore annihilates the $(2,0)$ generating subspace, and its conjugate,
and hence all of $W$ by Lemma~\ref{gw:hodge}. This proves vanishing as a
tensor, independently of the values of the two variables. The argument
also applies after projecting an expanded target to $Z$.
If the projected degree is zero, all evaluations in $Z$ agree. Factoring out
base classes by the projection formula, the two variable classes occur as
$u\smile v$. They already have top degree on the surface. Every additional
positive-degree base factor kills their product; the remaining fibre
integral is a rational number multiplying $\int_Zuv$. This reasoning is
independent of $u,v$ and allows the specified descendants.
\end{proof}
\begin{lemma}[A tree with two variables]\label{gw:path}
In either normal-cone degeneration above, consider a genus-zero tree with
exactly two variable insertions of
Hodge differences $+2$ and $-2$. Every branch disjoint from the path joining
them supplies a rational diagonal class at its attachment. Along that path
the variable spaces are copies of $T_{M,\Q}$ or $W$, with the appropriate
Tate shifts. Orient the path from the first labelled variable marking to
the second, and order the two slots at each path vertex in that direction.
Suppose that, with this slot order, each vertex tensor belongs to the
applicable row of the following table. Then contraction of the path
belongs to the row determined by its two remaining ends:
\[
\begin{array}{c|c}
\text{two end spaces}&\text{allowed tensors}\\ \hline
T_{M,\Q},T_{M,\Q}
& (u,v)\longmapsto q_M(Au,v),\quad A\in G_M,\\
W,W&\Q q_Z,\\
T_{M,\Q},W\text{ or }W,T_{M,\Q}&0.
\end{array}
\]
One may replace $G_M$ by any specified rational unital operator algebra,
provided that each path vertex still satisfies its applicable row in this
oriented slot order.
In the reconstruction arguments below, this vertex hypothesis is supplied
by the local-tensor lemma or by an earlier induction case.
\end{lemma}
\begin{proof}
Cut an edge outside the path. The detached connected subtree has only
rational diagonal input classes. Its single output is rational and of
Hodge difference zero, because its virtual class and all structure maps
respect Hodge difference. At the retained vertex the attachment remains
its existing relative root,
with the same contact and the resulting diagonal evaluation weight; it is
not an additional ordinary marking. Repeating this operation removes all
side branches from the tensor contraction. Conservation of Hodge difference
at the remaining vertices forces difference two along the path. Lemma~\ref{gw:hodge} and the
projective bundle formula identify the indicated variable spaces.
These path classes and the diagonal branch outputs have even cohomological
degree, so their permutations introduce no odd-degree signs.
For the first row, contraction of consecutive tensors written in path order
as $q_M(Au,x)$ and $q_M(By,v)$ pairs $x$ and $y$ with the inverse cup form
and gives $q_M(BAu,v)$. Thus contraction composes operators without taking
adjoints. In the second row it composes scalar maps between copies of $W$.
A path changing between the two sorts
has a mixed vertex, whose tensor vanishes by the stated vertex hypothesis.
The diagonal of a projective bundle gives
the same identities between its copies of $W$, with the relevant nonzero
pairing constants. The conclusion follows from closure under composition
and rational linear combination. If the two variables meet on one local
piece, Lemma~\ref{gw:local} supplies the second row directly.
\end{proof}
\subsection{Reconstruction with a center of codimension at least two}
We first prove a relative assertion for $(Y,D)$. Ordinary insertions are
pullbacks from $M$. Relative weights are written
\begin{equation}\label{gw:relative-weights}
\mu=\bigl((k_j,\xi^{b_j}p^*\delta_j)\bigr)_{j=1}^{L},\qquad k_j\geq1,
\quad0\leq b_j\leq r-1.
\end{equation}
The two variables may be ordinary $T_M$ insertions or relative $W$ insertions.
All other classes are rational and diagonal. A relative invariant with
these data is interpreted as a bilinear tensor on its two variable spaces.
Write $B_{\mathrm{rel}}(u,v)$ for such a tensor in numerical curve class
$\beta$ on $Y$, with relative list $\mu$. Admissibility requires
$D\cdot\beta=\sum_{j=1}^{L}k_j$.
The \emph{original ordinary markings} are the ordinary markings of this
relative datum, before translating any relative conditions into additional
absolute markings. This distinction fixes the ordinary-mark count used
in the induction.
The translation of relative conditions into absolute descendants follows
Hu--Li--Ruan \cite[Section~5.2, equation~(17)]{HLR08}, extending the divisor
correspondence of Maulik--Pandharipande
\cite[Theorem~2 and Section~2.4]{MP06}.
The induction below retains the two variable insertions and all original
ordinary descendants, in order to prove the required tensor restrictions.
\begin{lemma}[Restricted relative reconstruction]\label{gw:relative}
Every such tensor satisfies the three rows of Lemma~\ref{gw:path}.
\end{lemma}
\begin{proof}
Replace a relative weight $(k,\xi^bp^*\delta)$ by the ordinary insertion
\begin{equation}\label{gw:translation}
\tau_{r(k-1)+b}(i_*\delta)
\end{equation}
to form the absolute tensor $B_{\mathrm{abs}}(u,v)$ on $M$ in numerical
class $\pi_*\beta$, retaining all original ordinary insertions
and their descendant powers. We apply degeneration to the normal cone
using the following specific lifts. In
\[
\mathcal W=\operatorname{Bl}_{Z\times\{0\}}
(M\times\mathbb A^1)
\]
the strict transform $\mathcal Z$ of $Z\times\mathbb A^1$ is still
$Z\times\mathbb A^1$, because the blowup ideal restricts to the Cartier ideal
$(t)$. Lift $i_*\delta$ by $(i_{\mathcal Z})_*(\delta\times1)$.
On the special fibre its restriction to $Y$ is zero and its restriction to
$P=\mathbb P_Z(N_{Z/M}\oplus\mathcal O_Z)$ is the zero-section Gysin class.
These are transverse fibre restrictions. Original ordinary classes use
pullbacks from $M$.
This lift is valid even when $i_*\delta=0$. A class on $\mathcal W$ may
restrict to zero on the smooth fibre and nontrivially to its special
components. For a relative variable in $W$, Lemma~\ref{gw:hodge} says that
the associated absolute insertion is zero, so the absolute bilinear tensor
is zero. For two ordinary $T_M$ variables the associated absolute tensor
is a generator allowed in the first row. Thus the absolute starting tensor
has the required restriction in all three cases.
We prove the relative restriction by simultaneous strong induction, keeping
the two variables throughout. Fix an integral ample divisor $H$ on $M$.
The outer order is first $H\cdot\pi_*\beta$, then the number of original
ordinary markings. Within fixed outer data, use, in succession, the length
of the relative list, the sum of its contacts, and the sum of its
hyperplane exponents, with smaller values first. At every smaller outer
index the assertion is assumed simultaneously for all numerical curve
classes at that index, relative lists, evaluation labels, placements of
the two formal variable slots, and ordinary descendant exponents.
At equal outer indices the inner induction is likewise simultaneous in
the curve classes, labels, variable placements, and descendant exponents.
Cutting a side branch only replaces the weight
at an existing relative root, so it does not increase the number of original
ordinary markings.
A connected relative vertex on $Y$ of zero projected degree cannot have
joining markings. Indeed every nonzero effective curve contracted by
$\pi$ has class a positive multiple of an exceptional line, and has negative
$D$-degree; relative admissibility would require its $D$-degree to equal a
sum of positive contacts. A zero-class vertex likewise has no contacts.
Thus zero projected degree contributes only a constant invariant with no
joining marks. Its two middle-degree ordinary variable classes already
have total top degree; the invariant is a scalar cup pairing, or zero.
This supplies the initial case.
For a noninitial term of the degeneration formula, cut away the branches
with no variables, as in Lemma~\ref{gw:path}.
Every $Y$-vertex in a nontrivial connected tree has a joining edge, so the
zero-degree argument above makes its projected ample degree positive.
The projected classes of all components are effective and add to the
original class. Hence, if there is more than one $Y$-vertex, every one
has smaller projected degree and its path tensor is controlled by induction.
A unique $Y$-vertex of smaller projected degree is controlled in the same way.
If a $Y$-vertex has the full projected degree, it is the unique $Y$-vertex;
its tensor is still an earlier case if it retains fewer original ordinary
markings. Local path vertices are controlled by Lemma~\ref{gw:local}.
In these cases every path vertex therefore satisfies its row, so the
conditional contraction rule of Lemma~\ref{gw:path} controls the term.
A path containing no $Y$-vertex is controlled entirely by the local lemma.
It remains to consider a unique $Y$-vertex of the full projected class
carrying all original ordinary markings.
Every $P$-vertex is then a fibre tail with one joining marking, since the
graph is a tree. Let such a tail have joining contact $k'$ and receive the
translated markings indexed by a set $J$. All its evaluations in $Z$
agree. Its pushforward to the joining divisor therefore has the form
\[
\left(\prod_{j\in J}p^*\delta_j\right)\eta,
\]
where $\eta$ is a rational diagonal class independent of the base labels.
The projection formula gives this identity also when a label is one of the
two variables. The complex codimension of $\eta$ is
\begin{equation}\label{gw:tail-codim}
c=r\left(\sum_{j\in J}k_j-k'\right)
+\sum_{j\in J}b_j-|J|+1.
\end{equation}
Indeed the relative fibre virtual dimension is
$r-2+rk'+|J|$; each translated zero-section insertion has fibre codimension
$r+r(k_j-1)+b_j=rk_j+b_j$, and pushing to the relative fibre divisor of
dimension $r-1$ gives the displayed difference.
If one tail contains both variable markings, both are relative $W$
variables, since all original ordinary markings remain on the main vertex.
Their product is $q_Z(u,v)$ times the point class on $Z$, so the tail output
is $q_Z(u,v)$ times a fixed class. The remaining vertices then have only
fixed insertions, so the graph is a rational multiple of $q_Z$. It already
lies in the allowed $W,W$ row, without using induction on its main tensor.
For every other nonzero term, expand the tail outputs in the projective
bundle basis. A tail with one $W$ variable multiplies it only by fixed
diagonal classes, which act by scalars between the allowed copies or
annihilate it. Thus each remaining summand retains two separate variable
slots in the prescribed spaces, and the following induction applies to it.
A negative $c$ gives zero. An empty tail has $c=1-rk'<0$, so every tail
receives a translated marking. The new relative list therefore has no
more entries than the old one. A strict decrease is an earlier induction
case. If the lengths agree, each tail has exactly one translated marking.
Then $c\geq0$ and $b_j<r$ imply $k'\leq k_j$. A strict decrease of the
total contact is again earlier. At equal contacts, expand $\eta$ in the
projective bundle basis. Its fibre exponent is at most $b_j$, so either
the total hyperplane exponent decreases or all the original indices agree.
A decrease of a fibre exponent only multiplies its base label by a rational
diagonal class, and preserves the stated variable spaces. Thus all smaller
terms with separate variable slots retain the required tensor
restriction.
For one translated mark and equal indices, the coefficient of the highest
fibre power is obtained by restricting to a fibre. The relative invariant
computed in \cite[Theorem~7.1]{HLR08} is
\begin{equation}\label{gw:fibre-number}
\left\langle\tau_{rk-1-j}(\mathrm{pt})\mid H^j\right\rangle^%
{ (\mathbb P^r,\mathbb P^{r-1})}_{0,k[\mathrm{line}]}
=\frac{1}{k^{r-j}((k-1)!)^r}>0,
\qquad 0\leq j\leq r-1.
\end{equation}
The proof of that theorem identifies the descendant line with the
cotangent line of the source at the ordinary marked point, so its convention
is the one fixed above. Taking $j=r-1-b$ gives exactly the descendant in
\eqref{gw:translation}. Gluing multiplies these numbers by positive contact
and labeling factors. Consequently the remaining term is a nonzero
rational scalar times the original relative tensor. At fixed projection,
the total contact fixes the numerical class on the blowup, since the
kernel on curve classes is generated by the exceptional line.
We have an identity $B_{\mathrm{abs}}=C B_{\mathrm{rel}}+R$, with
$C\in\Q^*$ independent of the two variable vectors. The starting tensor
and the remainder belong to the relevant row of Lemma~\ref{gw:path}; hence
so does $B_{\mathrm{rel}}$. The induction is simultaneous in the three
rows. It is a tensor identity, not a separate choice of a scalar for each
pair of vectors. Testing on opposite off-diagonal types and using
Lemma~\ref{gw:hodge} extends it to the full rational transcendental
structures. This completes the induction.
\end{proof}
\subsection{The exceptional factor and recovery of the old algebra}
We first obtain
\begin{equation}\label{gw:inclusion}
G_Y\subseteq G_M\oplus\Q\operatorname{id}_{W}.
\end{equation}
Degenerate $Y$ along its divisor $D$. The main component is again $Y$,
and the other component is the projective-line bundle $P_D\to D$ defined above.
By the blowup cohomology formula every ordinary insertion on $Y$ is a sum
of a pullback from $M$ and exceptional terms $j_*a$, where $j:D\hookrightarrow
Y$. Lift an exceptional term by the strict transform of the divisor family,
so that it is supported on the zero section of the bundle component.
Use pullbacks for the other terms. Thus the ordinary insertions on the
main relative component are exactly those of Lemma~\ref{gw:relative}.
On the other components the local rule of Lemma~\ref{gw:local} applies,
through their projection to $Z$. Applying Lemma~\ref{gw:path} to each
degeneration tree proves \eqref{gw:inclusion}, including zero mixed blocks.
The inclusion controls every operator but does not yet separate the two
factors or recover all operators on the old summand. The exceptional
line gives the required separation.
If $W\neq0$, let $e$ be the exceptional line class. Every stable map of
class $e$ is contained in one fibre of $D\to Z$. A fibre has normal bundle
$\mathcal O(-1)\oplus\mathcal O^{\oplus2}$ in $Y$, so these maps are
unobstructed. The two-mark degree-one evaluation map on the fibre has
degree one onto $\mathbb P^1\times\mathbb P^1$. Restricting
$j_*p^*u$ to $D$ gives $-\xi p^*u$, whence
\begin{equation}\label{gw:exceptional-line}
\left\langle j_*p^*u,j_*p^*v\right\rangle^Y_{0,e}
=\int_Zuv.
\end{equation}
An insertion from $T_M$ restricts to zero on $D$ by
Lemma~\ref{gw:hodge}. Relative to the negative cup form of the exceptional
summand, \eqref{gw:exceptional-line} is minus its projector. Thus the
projector onto $W$ belongs to $G_Y$. Its complement belongs to $G_Y$ too.
It remains to recover $G_M$. For this purpose we need the divisor version
of Lemma~\ref{gw:relative}, now allowing all ordinary classes on $Y$.
Write $\operatorname{pr}_M G_Y$ for its algebra of restrictions to $T_M$;
this is defined by \eqref{gw:inclusion}.
\begin{lemma}[Divisor reconstruction with separated variable spaces]
\label{gw:divisor}
For $(Y,D)$, allow arbitrary ordinary diagonal insertions and arbitrary
ordinary descendants. Allow the two variables either as ordinary classes
in $T_M$ or $W\subset H^4(Y)$, or as boundary classes in a tautological
copy of $W$ in $H^*(D)$. Then relative tensors with two $T_M$ ends belong
to $\operatorname{pr}_M G_Y$, tensors with two $W$ ends are scalar cup
pairings, and mixed tensors vanish.
\end{lemma}
\begin{proof}
Fix a relative tensor $B_{\mathrm{rel}}(u,v)$ in numerical curve class
$\beta$ on $Y$, with relative list
$\mu=((k_j,\delta_j))_{j=1}^{L}$. Thus
$D\cdot\beta=\sum_{j=1}^{L}k_j$.
Use degeneration of $Y$ along $D$, with the same source cotangent convention,
and translate a relative weight $(k,\delta)$ into
$\tau_{k-1}(j_*\delta)$ to form an absolute tensor
$B_{\mathrm{abs}}(u,v)$ on $Y$ in the same numerical class $\beta$,
retaining the original ordinary insertions and descendants.
Use the strict divisor-family lift for translated
classes and pullbacks for original ordinary classes. This is the
classical divisor correspondence of \cite[Section~2.4, Lemma~4]{MP06}; we
give the ordering needed for the restricted tensors.
The ordinary $T_M$ part restricts to zero on $D$. The rational Hodge
structures generated by boundary classes of Hodge difference $\pm2$ are
copies of $W$: for a surface center they are
$p^*W\subset H^2(D)$ and $\xi p^*W\subset H^4(D)$.
Gysin carries the first into the exceptional $W$ summand of $H^4(Y)$ and
annihilates the second, since $H^6(Y)$ is diagonal. Restriction of the
ordinary exceptional class is $j^*j_*p^*u=-\xi p^*u$.
It follows from \eqref{gw:inclusion} that every associated absolute tensor
has the claimed restriction. Local tensors have it by
Lemma~\ref{gw:local}. This also treats a translated variable with zero
Gysin class; the absolute tensor is then zero.
Fix an integral ample divisor on $Y$ and perform outer strong induction
on its curve degree, then on the number of original ordinary markings.
As before, these are the ordinary markings before translation, and the
induction is simultaneous for all numerical classes, relative data,
labels, variable placements, and descendant exponents at the given index.
Pruning replaces existing relative weights as in Lemma~\ref{gw:path}.
Every main $Y$-vertex in a nontrivial tree has a joining root. Its curve
class cannot be zero, since a zero class has total contact zero; its ample
degree is therefore positive. Under collapse of the degeneration to $Y$,
the projected component classes are effective and add to the original
class $\beta$. If there are several main vertices, all have smaller ample
degree. A unique main vertex of smaller degree is also an earlier case.
A main vertex with full degree is unique, and is an earlier case
unless it retains all original ordinary markings. Thus in these cases
induction controls every main path vertex and Lemma~\ref{gw:local}
controls every local one, as required by the conditional path rule.
A path with no main vertex is controlled by the local lemma after pruning.
At equal outer data the unique main vertex has curve class $\beta$:
all other components project to points of $Y$. They are fibre tails of
$P_D\to D$, since the graph is a tree. This argument uses the ample degree
on $Y$, without requiring $D$ to be nef. Hence
$K=D\cdot\beta$, the total contact, is fixed. If $K<0$ there is no
admissible relative datum. A zero curve class has $K=0$ and no relative
marks; its constant tensors are scalar cup pairings, zero on mixed
summands. More generally, for $K=0$ the following inner induction has just
the empty-list case.
Let the new main relative list have length $t$, with tail contacts $k'_a$.
If $J_a$ is the set of
translated markings on tail $a$, the tail calculation
\eqref{gw:tail-codim} now applies with $r=1$: the center of this
normal-cone degeneration is the divisor $D\subset Y$.
Thus
\begin{equation}\label{gw:divisor-order}
c_a=\sum_{j\in J_a}(k_j-1)-k'_a+1,
\qquad
\sum_a c_a=t-L.
\end{equation}
Before applying the list order, use the tail-output factorization from the
first reconstruction. A tail containing both variable slots gives
$q_Z(u,v)$ times fixed classes, so its whole graph contribution is already
an allowed scalar pairing. In every other nonzero summand the variable
slots remain separate: fixed diagonal classes act by scalars between the
copies of $W$. The following inner induction applies to these summands.
Each nonzero tail has $c_a\geq0$. Therefore $t\geq L$.
Use decreasing length as the next order; it terminates because
$t\leq K$. This is an inner induction for each fixed $\beta$, so no
unbounded reversal of an order is being used.
At equal lengths every $c_a$ is zero. Each tail has
\[
k'_a-1=\sum_{j\in J_a}(k_j-1),\qquad
\delta'_a=\text{a scalar times }\prod_{j\in J_a}\delta_j.
\]
Take all fixed labels homogeneous. Give $(k_j,\delta_j)$ weight
$k_j-1+\operatorname{codim}\delta_j$. These weights are nonnegative.
Each formal variable slot has a fixed cohomological degree, unchanged
when its vector is specialized to zero or to the other vector. A boundary
$W$ slot has positive codimension, so it always has positive weight.
Combining two positive-weight labels on a tail decreases their number, so
use that number, with smaller values first, as the final order.
A strict decrease of this count is an earlier case with two separately
labelled variable slots.
If the number of positive-weight labels is unchanged, each tail has at most
one positive-weight label. All other labels are $(1,1)$, so every nontrivial contact and label
is unchanged. Empty tails have degree one and identity output; redistribution
of the trivial labels, at the fixed total length, consequently gives the
original weighted partition, with each formal variable slot still
distinguished.
These unchanged-data terms cannot cancel. A tail carrying its one
positive-weight label has the leading fibre invariant in
\eqref{gw:fibre-number}, now for $r=1$, with its positive gluing contact
factor. An additional $(1,1)$ label translates to an ordinary point-divisor
insertion on $\mathbb P^1$. By the divisor equation it multiplies the
leading number by $k'_a$. The descendant correction contains the square
of the fibre point class and vanishes. For the source cotangent convention
this is the ordinary relative divisor equation with the divisor supported
in the interior, disjoint from the relative boundary. A degree-one tail
without translated markings contributes one. Thus every unchanged-data
coefficient is a product of the same positive leading numbers and positive
contact, divisor, and labeling factors. All path labels are even, so no
permutation signs occur. The graph with one leading fibre tail for each
relative entry is among these terms. Their sum is therefore a nonzero
rational coefficient $C$, depending only on the fixed discrete data and
the formal labels, not on the two variable vectors. For the empty list
the coefficient is one. This is the nonvanishing part of the triangular
coefficient calculation in \cite[Section~2.4, Lemma~4]{MP06}; its exact
normalization is not needed here.
As before, the formula is an identity of bilinear tensors. All earlier
terms are allowed by the simultaneous induction and the path lemma; division
by the nonzero, vector-independent coefficient proves the three stated
restrictions. Ordinary descendants at retained markings have not changed,
and those at markings moved to a fibre occur only in the smaller ordinary-mark
cases. Hence no bound or extra hypothesis on original descendants or
ordinary diagonal insertions has been used.
\end{proof}
To finish Theorem~\ref{gw:blowup}, degenerate any absolute generator on $M$
along $Z$. Its two $T_M$ insertions have zero restriction to the normal
bundle component, so they occur on the main side. By
Lemma~\ref{gw:divisor}, a main vertex carrying both insertions supplies
a tensor in $\operatorname{pr}_M G_Y$ after its side branches are pruned.
If the two insertions lie on different main vertices, the path leaves
the first through a boundary copy of $W$. That vertex has one $T_M$ end
and one $W$ end, so its tensor vanishes by the mixed row of the same lemma.
When $W=0$, there is no Hodge-difference-two boundary class through which
such a path could pass. Local tensors and branch outputs are controlled
by Lemmas~\ref{gw:local} and~\ref{gw:path}. Thus each term, and hence
this absolute generator, belongs to
$\operatorname{pr}_M G_Y$. Together with \eqref{gw:inclusion} this gives
$G_M=\operatorname{pr}_M G_Y$. The exceptional-line projector separates
the other factor when it exists. This proves the theorem.
\subsection{Integral cancellation and the surface forced by rationality}
\begin{lemma}[Whole blocks]\label{gw:blocks}
Along a smooth weak factorization starting at $\mathbb P^4$, each
transcendental lattice is an integral orthogonal sum of whole shifted
surface-transcendental lattices. Its operator algebra is the product of
independent rational scalar algebras on those summands. A blowdown with
nonzero surface-transcendental contribution removes one whole summand,
whose integral polarized Hodge-isometry class is that of the entire
transcendental lattice of its center. A step with zero transcendental
contribution leaves the decomposition unchanged.
\end{lemma}
\begin{proof}
At $\mathbb P^4$ the transcendental lattice is zero. A blowup with a
nonzero surface contribution adds exactly one summand by
Theorem~\ref{gw:blowup}; a blowup with no transcendental contribution does
nothing. A blowdown with zero transcendental contribution also leaves the
lattice and algebra unchanged by the same theorem. Suppose instead that
the next step is a blowdown with nonzero surface contribution. The theorem,
read in reverse, exhibits the removed center as a factor $\Q$ of
the operator algebra of the current variety. Its identity is a primitive
central idempotent. The primitive central idempotents of the already
identified algebra $\Q^m$ are precisely the projectors onto its existing
whole summands. Thus the removed rational subspace is one of those
summands, even if some summands are mutually Hodge-isomorphic or a surface
transcendental structure is reducible.
The lattice of each summand is its intersection with the ambient integral
lattice: the historical decomposition and the geometric blowup decomposition
are integral direct sums. Equality of their rational subspaces therefore
identifies their saturated integral lattices and cup forms. The residual
summands are unchanged. This proves the induction.
\end{proof}
The numerical comparison below is the rank-twenty-one counterpart of
the surface-classification argument in \cite[Lemma~3]{Kulikov08}.
\begin{lemma}[Rank twenty-one centers]\label{gw:k3-center}
Let $Z$ be a smooth connected projective surface with
$h^{2,0}(Z)=1$ and $\operatorname{rank}T_Z=21$. Then $Z$ is birational to a
Picard-number-one K3 surface. Its primitive ample generator has square
$|\operatorname{disc}T_Z|$. If this square is $d>2$ and
$\operatorname{End}_{\mathrm{Hdg}}(T_{Z,\Q})=\Q$, that K3 surface has
trivial automorphism group.
\end{lemma}
\begin{proof}
Geometric genus and the transcendental lattice are unchanged by point
blowups. Pass to a minimal model using \cite[Proposition~II.16]{Beauville78}.
In the following numerical calculation, $K$, $b_2$, and $q=h^{1,0}$
refer to that minimal surface.
Since $p_g=1$, it is not ruled and has nonnegative Kodaira dimension
\cite[Propositions~III.20--III.21 and Example~VII.3]{Beauville78}. Its canonical class is nef
\cite[Corollary~VI.18]{Beauville78}, so $K^2\geq0$.
Noether's formula \cite[Sections~I.14 and~III.19]{Beauville78} gives
$c_2=12(1-q+p_g)-K^2=24-12q-K^2$.
On the other hand its topological Euler characteristic is
$c_2=2-4q+b_2$. Hence
\[
b_2=22-8q-K^2.
\]
Projectivity gives Picard rank at least one, while the rank assumption gives
$b_2\geq22$. Thus $q=0$, $K^2=0$, and the Picard rank is one.
Kodaira dimension two is excluded by $K^2=0$ \cite[Proposition~X.1]{Beauville78}. In Kodaira dimension one the
elliptic fibration \cite[Proposition~IX.2]{Beauville78} supplies a
nonzero isotropic divisor class, independent
of an ample class, contradicting Picard rank one. The classification in
Kodaira dimension zero leaves a K3 surface, since $p_g=1$ and $q=0$
\cite[Theorem~VIII.2]{Beauville78}.
The K3 lattice is unimodular, and its N\'eron--Severi lattice is primitive
by the integral Lefschetz $(1,1)$ theorem
\cite[Theorem~2]{VoisinHodge2016}.
For a primitive sublattice of a unimodular lattice, its discriminant group
and that of its orthogonal have equal orders. The N\'eron--Severi lattice
here is generated by a primitive ample class $L$, so its discriminant is
$L^2$. This proves the degree assertion.
Any automorphism fixes $L$ and hence acts trivially on this rank-one
N\'eron--Severi lattice. Its action on $T_{Z,\Q}$ is a rational scalar by
the endomorphism hypothesis, and preservation of the nondegenerate form
makes that scalar $+1$ or $-1$. The unimodular gluing identifies the
cyclic discriminant group of $T_Z$ with that of $\Z L$, of order $d$;
see \cite[Proposition~1.6.1]{Nikulin80} for the compatible gluing of
primitive orthogonal sublattices of an even unimodular lattice.
The action $-1$ on the first and $+1$ on the second is compatible only if
$2$ annihilates this cyclic group, contrary to $d>2$. Thus the action is
$+1$ on both lattices, and hence on the whole integral second cohomology.
The faithful cohomological action for K3 automorphisms, a consequence of
global Torelli \cite[Proposition~15.2.1]{Huybrechts16}, makes the
automorphism the identity.
\end{proof}
\begin{proposition}[The contribution forced by rationality]
\label{gw:net-center}
Let $M$ be a rationally connected smooth projective fourfold with
\[
h^{3,1}(M)=1,\qquad \operatorname{rank}T_M=21,\qquad
|\operatorname{disc}T_M|=d>2,\qquad
\operatorname{End}_{\mathrm{Hdg}}(T_{M,\Q})=\Q.
\]
Let $\mathcal S$ consist of the Picard-number-one K3 surfaces $S$ whose
shifted integral transcendental lattice $(T_S(-1),-q_S)$ is Hodge-isometric
to $(T_M,q_M)$, and use this collection in the test $u$ of
the preliminary section. If $f:\mathbb P^4\dashrightarrow M$ is a
birational map, then
\[
c_u(f)=1.
\]
Every member of $\mathcal S$ has degree $d$ and trivial automorphism group.
\end{proposition}
\begin{proof}
Under the rationality assumption all varieties in the smooth factorization
of Theorem~\ref{pre:weak-factorization} are rationally connected.
Lemma~\ref{gw:blocks} tracks integral whole blocks through the factorization.
A nonzero transcendental rational Hodge structure of this kind has a
nonzero off-diagonal part by Lemma~\ref{gw:hodge}. Since the final
$h^{3,1}$ is one, its transcendental structure is simple: a nontrivial
semisimple decomposition would require at least two nonzero $(3,1)$ parts.
Thus the final lattice is one whole block. The number of additions minus
removals of blocks in its integral polarized Hodge-isometry class is one.
Every center in this class has geometric genus one, transcendental rank
$21$, determinant of absolute value $d$, and rational Hodge endomorphism
ring $\Q$. Lemma~\ref{gw:k3-center} makes it birational to a member of
$\mathcal S$, and proves the degree and automorphism assertions. Conversely,
a smooth surface center counted by $u$ is birational to a member of
$\mathcal S$: in dimension two an MRC base which is a K3 has the same
function field as the surface. Its entire transcendental lattice therefore
has exactly the required class. Point and curve centers are not counted.
The exceptional divisor of a smooth blowup is a projective bundle over its
center and has the same MRC base, by Lemma~\ref{pre:mrc-fibration}.
The sign convention and additivity in Lemma~\ref{pre:cocycle}
therefore identify $c_u(f)$ with the above number of block additions minus
removals. This number is one.
\end{proof}
==== END Irrational-cubic-fourfolds-with-geometric-K3-categories//gw.tex ====
==== BEGIN Irrational-cubic-fourfolds-with-geometric-K3-categories//introduction.tex ====
\section{Introduction}\label{sec:introduction}
A smooth cubic fourfold \(X\subset\mathbf P^5_{\C}\) is rational if its field of rational functions is
a purely transcendental extension of \(\C\) of transcendence degree four. The connection
between this birational question and K3 surfaces appears both in the
middle cohomology and in the derived category. We prove that both
connections can hold for the same irrational cubic.
The geometric connection grew out of explicit rationality constructions.
Cubics containing quartic rational scrolls were studied by Fano
\cite[p.~72]{Fano43}; Tregub gave constructions using two disjoint
planes, quintic del Pezzo surfaces, and quartic scrolls
\cite[Sections~2--4]{Tregub84}.
Beauville and Donagi proved that the Fano variety \(F(X)\), which
parametrizes lines on \(X\), is irreducible holomorphic symplectic.
For sufficiently general Pfaffian cubics they identified \(F(X)\) with
\(S^{[2]}\), the variety of length-two subschemes of a K3 surface \(S\),
and proved that the cubic is rational
\cite[Propositions~1--2 and~5]{BeauvilleDonagi85}.
Hassett constructed further rational cubics containing a plane; in those
parametrizations a family of exceptional lines is parametrized by a
surface birational to a K3 surface
\cite[Theorem~4.2 and Proposition~5.1]{Hassett99}.
The Kuznetsov component of \(X\) is the full triangulated subcategory
\[
\Ku(X)=\bigl\{F\in D^b(\operatorname{Coh}X):
\Hom(\mathcal O_X(i),F[k])=0
\text{ for }0\le i\le2\text{ and }k\in\Z\bigr\}.
\]
It gives the semiorthogonal decomposition
\[
D^b(\operatorname{Coh}X)
=\langle\Ku(X),\mathcal O_X,\mathcal O_X(1),\mathcal O_X(2)\rangle.
\]
Kuznetsov's rationality conjecture asserts that \(X\) is rational if and only
if \(\Ku(X)\) is equivalent to the derived category of a projective K3
surface \cite[Conjecture~1.1]{Kuznetsov10}. Here rationality means
birationality over \(\C\) with \(\mathbf P^4\), and the K3 equivalence in
this statement means an exact \(\C\)-linear equivalence with the ordinary category
\(D^b(\operatorname{Coh}S)\). The target has no Brauer twisting.
Let \(\mathcal C\) denote the moduli space of smooth complex cubic
fourfolds. Write \(h=c_1(\mathcal O_X(1))\). A \emph{labelling of
discriminant \(d\)} is a saturated positive-definite rank-two sublattice
\[
K\subset H^4(X,\Z)\cap H^{2,2}(X)
\]
containing \(h^2\), with \(\det K=d\). The corresponding Hassett locus is
denoted by \(\mathcal C_d\). We call an integer \(d\) \emph{admissible} if
\begin{equation}\label{cub:admissible}
d>6,\qquad d\equiv0,2\pmod6,\qquad
4\nmid d,\quad 9\nmid d,\quad
p\nmid d\ \text{for every odd prime }p\equiv2\pmod3.
\end{equation}
Hassett's period and lattice theorems make \(\mathcal C_d\) a nonempty
irreducible divisor for these discriminants and give its associated
polarized K3 surfaces \cite[Theorems~1.0.1 and~1.0.2]{Hassett00}.
Here an associated K3 surface has a primitive polarization \(L\) of
degree \(d\) and an integral Hodge isometry
\(K^\perp\simeq L^\perp(-1)\), where the surface cup pairing is reversed.
Kuznetsov verified the categorical prediction for smooth Pfaffian cubics
\cite[Theorem~3.1]{Kuznetsov10}. Addington and Thomas proved that the
Hodge-theoretic and categorical K3 associations agree on a dense open
subset of each admissible divisor \cite[Theorem~1.1]{AddingtonThomas14}.
The construction of stability conditions by Bayer, Lahoz, Macr\`i and
Stellari \cite[Theorem~1.2]{BLMS23}, together with the theory in families
of Bayer, Lahoz, Macr\`i, Nuer, Perry and Stellari, gives an ordinary K3
derived category for every cubic having an associated K3 surface
\cite[Corollary~29.7]{BLMPNS21}. These results supply the categorical
assertion below; the irrationality assertion is proved here.
Our main result disproves the categorical-to-rational implication of
Kuznetsov's conjecture on every sufficiently large admissible divisor.
\begin{theorem}[Irrationality on large admissible Hassett divisors]\label{thm:main}
There is an integer \(d_0\) such that, for every admissible \(d>d_0\), a very
general cubic fourfold \(X\) in \(\mathcal C_d\) is irrational
and admits an exact \(\C\)-linear equivalence
\[
\Ku(X)\simeq D^b(\operatorname{Coh}S)
\]
for a smooth projective K3 surface \(S\).
\end{theorem}
The threshold \(d_0\) is ineffective. Some Hassett divisors consist
entirely of rational cubics: Russo and Staglian\`o proved rationality throughout
\(\mathcal C_{26}\) and \(\mathcal C_{38}\) using congruences of
five-secant conics \cite[Theorems~4 and~7]{RussoStagliano19}, and
throughout \(\mathcal C_{42}\) using trisecant flops
\cite[Theorem~5.12]{RussoStagliano23}.
Here ``very general'' means outside a
countable union of proper closed algebraic subsets separately in each
divisor \(\mathcal C_d\).
This qualification matters: Yang and Yu construct rational subloci of
codimension two in the full moduli space inside every Hassett divisor
\cite[Theorem~9]{YangYu20}.
\subsection{The two evaluations of one divisorial invariant}
Write \(T_X\) for the integral lattice orthogonal to the integral
Hodge classes in \(H^4(X,\Z)\), with its cup-product pairing, and put
\(d=|\det T_X|\). The irrationality argument concerns cubics with
\(\operatorname{rank}T_X=21\), \(d>2\), and
\(\operatorname{End}_{\mathrm{Hdg}}(T_X\otimes\Q)=\Q\).
Suppose such an \(X\) were rational, and choose a birational map
\(f:\mathbf P^4\dashrightarrow X\).
We use one fixed collection \(\mathcal S\) of
Picard-number-one K3 surfaces: those whose entire integral
transcendental Hodge lattice, after shifting Hodge types by \((1,1)\)
and reversing the pairing, is isometric to \(T_X\).
The lattice hypotheses imply that these surfaces have primitive ample
degree \(d\) and trivial automorphism group
(Lemma~\ref{gw:k3-center}). We do not assume that this collection is
nonempty: the hypothetical birational map will force a member of it.
The maximal rationally connected (MRC) quotient of a smooth projective
variety is the base of its birationally determined rational fibration with
rationally connected general fibres and non-uniruled base.
For an integral variety \(E\), let \(u(E)\) be one if the MRC quotient
of a smooth projective model of its function field is birational to a
member of \(\mathcal S\), and zero otherwise. Section~\ref{pre:section} defines \(c_u(f)\) by counting the
prime divisors acquired by the target with positive sign and those
lost from the source with negative sign, each weighted by \(u\).
The count is finite and additive under composition. We evaluate this
same invariant in two ways.
Smooth weak factorization expresses \(f\) as blowups and blowdowns
along smooth centers. Only surface centers can have nonzero test.
Classically, a surface blowup adds its whole transcendental lattice,
shifted to weight four with reversed pairing, as an integral orthogonal
summand. The difficulty is to recognize that same summand on blowdown.
Section~\ref{gw:section} uses genus-zero Gromov--Witten pairings with
two variable transcendental inputs and all remaining insertions rational
of diagonal Hodge type. For an intermediate fourfold \(M\),
define \(T_M\) as for \(X\). Each such pairing \(B(a,b)\) determines an
operator \(A_B\) by
\(B(a,b)=q_M(A_Ba,b)\), where \(q_M\) is the cup form on
\(T_M\otimes\Q\). The algebra generated by these operators has a
separate scalar factor on each whole surface contribution. Its
projectors force a later blowdown to remove a whole block. Intersecting
that rational block with the ambient integral lattice recovers its
entire integral lattice, so cancellation preserves the exact test
class. Every nonzero block has a nonzero \((3,1)\) part, so
\(h^{3,1}(X)=1\) leaves one block in the final transcendental lattice.
Rank \(21\) makes its surface center birational to a
Picard-number-one K3 surface, and its discriminant fixes the degree
\(d\). Thus one more member of the class \(\mathcal S\) is added than
removed, and Proposition~\ref{gw:net-center} gives \(c_u(f)=1\).
The Sarkisov factorization passes through Mori fibre spaces: contractions
\(V\to B\) with terminal \(\Q\)-factorial projective fourfold \(V\),
\(\dim B<4\), relative Picard number one, and relatively ample
\(-K_V\). These are the nodes of the factorization. An elementary
birational transformation between nodes is a link; its endpoint bases
map to a common base \(Q\). The bases can change along the path.
Sections~\ref{bd:section}--\ref{sf:section} construct integer-valued
functions \(v,s\) on the nodes and prove, for sufficiently large \(d\),
\[
c_u(\text{link})=(v+s)(V'/B')-(v+s)(V/B).
\]
The value \(v(V/B)\) is the finite cancelled difference between the
tests of vertical divisors on \(V\) and divisors on \(B\).
The value \(s\) comes from the remaining generic-surface calculation.
At a conic-bundle node, the base field \(L=\C(B)\) has transcendence
degree three. For a surface subfield \(k\subset L\) arising from a link,
\(L\) is the function field of a genus-zero curve over \(k\). The surface
calculation
then gives a candidate for \(s\), using the test on finite residue
fields and covers determined by the conic's Brauer ramification.
The crucial point is that the candidates agree for every such
presentation with the ramification bound supplied by the links.
A second Sarkisov factorization, now on the threefold bases, compares
these presentations. Bounded diagrams with their branch loci recorded
give genus bounds incompatible with pencils on Picard-number-one K3
surfaces of sufficiently large degree. This proves independence with
one bound for all test subcollections and factorizations. At nodes over
a point both corrections vanish; the displayed differences therefore
telescope to \(c_u(f)=0\) by Theorem~\ref{sf:vanishing}.
The contradiction proves the irrationality criterion with one threshold
for all the lattice data in its statement. Section~\ref{cub:section} then
applies the standard period and categorical results: for each admissible
\(d\), a very general point of \(\mathcal C_d\) has the required lattice
data and an exact \(\C\)-linear equivalence of its Kuznetsov component
with the ordinary derived category of a smooth projective K3 surface.
The criterion therefore applies in every admissible divisor above the
same threshold. Figure~\ref{fig:two-evaluations} summarizes the
contradictory evaluations of the hypothetical map.
\begin{figure}[ht]
\centering
\begin{tikzcd}[column sep=large,row sep=large]
& \mathbf P^4\dashrightarrow X
\arrow[dl,"\text{smooth factorization}" description]
\arrow[dr,"\text{Mori factorization}" description] & \\
\text{whole integral surface blocks}
\arrow[d,"\text{Proposition~\ref{gw:net-center}}" description]
&& \text{endpoint corrections }v+s
\arrow[d,"\text{Theorem~\ref{sf:vanishing}}" description] \\
c_u(f)=1 && c_u(f)=0
\end{tikzcd}
\caption{Assuming rationality gives a birational map
\(f:\mathbf P^4\dashrightarrow X\). Its two evaluations use the same
collection of K3 surfaces.
The right-hand evaluation holds above a degree threshold independent
of the map, the test subcollection, and the number of links.}
\label{fig:two-evaluations}
\end{figure}
\FloatBarrier
\subsection{Consequences for K3 geometry and zero-cycles}
The modern associated-K3 rationality prediction, in the explicit
formulation recorded by Huybrechts
\cite[Conjecture~2.1, equation~(2.1)]{HuybrechtsUpdate25}, asserts that a
smooth complex cubic fourfold is rational if and only if the primitive
cohomology of some polarized K3 surface admits a primitive isometric Hodge
embedding into its primitive fourth cohomology, with the Tate twist and
pairing convention used below.
\begin{corollary}[Failure of associated-K3 sufficiency]\label{cor:associated-k3}
Let \(d_0\) be the threshold in Theorem~\ref{thm:main}. For every admissible
\(d>d_0\), a very general \(X\in\mathcal C_d\) is irrational and admits a
labelling \(K\) of discriminant \(d\) and a primitively polarized smooth
projective K3 surface \((S_{\mathrm{Hdg}},L)\), with \(L^2=d\), for which
there is an integral Hodge isometry
\[
K^\perp \simeq L^\perp(-1).
\]
Here \(L^\perp\subset H^2(S_{\mathrm{Hdg}},\Z)\), and the surface cup
pairing is reversed. Thus an associated untwisted polarized K3 surface in
this Hodge-theoretic sense is not sufficient for rationality.
\end{corollary}
The proof is given in Section~\ref{cub:consequences}.
\begin{corollary}[Failure of Hilbert-square sufficiency]\label{cor:hilbert-square}
Let \(d_0\) be the threshold in Theorem~\ref{thm:main}. For every integer
\(n\ge2\) such that \(d=2n^2+2n+2>d_0\), a very general
\(X\in\mathcal C_d\) is irrational and its Fano variety of lines \(F(X)\)
is birational to the Hilbert square \(S_{\mathrm{Hilb}}^{[2]}\) of a smooth
projective K3 surface \(S_{\mathrm{Hilb}}\). There are infinitely many such
discriminants. Thus birationality of \(F(X)\) to a K3 Hilbert square is
not sufficient for rationality of \(X\).
\end{corollary}
The proof is given in Section~\ref{cub:consequences}.
The categorical surface \(S\), the Hodge-associated surface
\(S_{\mathrm{Hdg}}\), and, in the Hilbert-square subfamily, the surface
\(S_{\mathrm{Hilb}}\) need not be identified. They witness different
structures on the same cubic.
\begin{corollary}[Integral Chow decompositions for the irrational cubics]
\label{cor:chow-diagonal}
Let \(d_0\) be the threshold in Theorem~\ref{thm:main}. For every admissible
\(d>d_0\), a very general \(X\in\mathcal C_d\) is irrational and has
universally trivial \(\operatorname{CH}_0\): for every field extension
\(F/\C\), the degree map
\[
\deg\colon\operatorname{CH}_0(X_F)\longrightarrow\Z
\]
is an isomorphism. This universal \(\operatorname{CH}_0\) property is
equivalent to the existence of a point \(x\in X(\C)\), a proper closed
algebraic subset \(D\subsetneq X\), and a codimension-four cycle \(\Gamma\)
with integer coefficients supported on \(D\times X\) such that
\[
[\Delta_X]=[X\times\{x\}]+[\Gamma]
\quad\text{in }\operatorname{CH}^4(X\times X)
\]
with integral coefficients.
\end{corollary}
The proof is given in Section~\ref{cub:consequences}.
This applies in particular to the associated-K3 examples in
Corollary~\ref{cor:associated-k3} and the Hilbert-square subfamily in
Corollary~\ref{cor:hilbert-square}. The asserted identity is an integral
Chow-theoretic decomposition, not merely a rational cohomological
decomposition; thus the decomposition-of-the-diagonal obstruction does not
detect the irrationality of these examples. A variety is stably rational
if its product with some projective space is rational. Stable rationality
implies universal \(\operatorname{CH}_0\)-triviality \cite{Voisin17}; the
corollary supplies this necessary condition for the irrational cubics.
No assertion about stable rationality is made.
\subsection{Relation to birational invariants and quantum methods}
The existence of an associated K3 is distinct from the stronger condition
that the Fano variety of lines be birational to a Hilbert square of a K3
surface \cite[Theorems~1 and~2]{Addington16}.
A related motivic approach uses the identity of Galkin and Shinder
between a cubic, its Hilbert square and its Fano variety of lines in the
Grothendieck ring \cite[Theorem~5.1]{GalkinShinder14}.
Their rationality consequence assumes a cancellation conjecture for the
class of the affine line \cite[Conjecture~2.7 and Theorem~7.5]{GalkinShinder14};
Borisov proved that this global cancellation conjecture fails
\cite[Theorem~2.12]{Borisov18}.
Our argument uses an additive invariant of individual birational maps.
Quantum cohomology and Hodge theory provide another approach to
irrationality. Iritani's blowup theorem decomposes the quantum
\(D\)-module after a formal extension of coefficients and change of
variables \cite[Theorem~1.1]{Iritani25}. Using this quantum blowup
behavior, Katzarkov, Kontsevich, Pantev and Yu construct Hodge atoms
and obtain an irrationality theorem for a very general cubic in the full
moduli space \cite[Theorem~6.8]{KKPY26}. Their very-general hypothesis
does not cover a prescribed Hassett divisor. Gu\'er\'e obtains a
rational Hodge-theoretic constraint on a rational cubic
\cite[Theorem~63]{Guere26}; that conclusion does not assert an integral
isometry of the intersection lattices.
The argument here retains whole integral polarized transcendental
lattices through cancellation. Their integral discriminants determine
the exact degrees of the K3 surfaces in the fixed test collection.
The integral comparison and the uniform bound for the relevant surface
contributions are proved in Sections~\ref{gw:section}--\ref{sf:section}.
Integral surface-transcendental lattices also enter Kulikov's approach,
which derives irrationality of a very general cubic fourfold from an
indecomposability conjecture for those lattices
\cite[Proposition~1]{Kulikov08}. Auel, B\"ohning and Graf von Bothmer
disproved that conjecture for the sextic Fermat surface
\cite[Theorem~1.2]{AuelBohningBothmer13}.
Here the independent factors of the operator algebra distinguish whole
surface contributions even when their transcendental Hodge structures
are reducible; no general indecomposability assertion is used.
Fay proves that a very general member of a special divisor is irrational
when the quaternion class \((d/2,-3)\) over \(\Q\) is nonzero
\cite[Theorem~1.1]{Fay26}.
The class vanishes exactly when the valuation of \(d/2\) is even
at every prime congruent to two modulo three
\cite[condition~\((**')\)]{Fay26}.
For every admissible \(d\), all these valuations are zero: \(4\nmid d\)
makes \(d/2\) odd, and admissibility excludes the other such primes.
Thus Fay's quaternion obstruction vanishes throughout the range of
Theorem~\ref{thm:main}.
The divisorial invariant is a specialization of the motivic invariants of
birational maps studied by Lin and Shinder \cite{LinShinder24}.
Their later work develops horizontal and vertical versions
\cite[Definition~2.3 and Lemmas~2.4--2.5]{LinShinder26} and constructs
unbounded contributions from elliptic surfaces
\cite[Theorem~1.2 and Section~4]{LinShinder26}.
The surface calculation adapts the virtual
N\'eron--Severi viewpoint of Lin, Shinder and Zimmermann
\cite[Definition~5.2 and Proposition~5.5]{LinShinderZimmermann23}.
Our uniform exclusion concerns Picard-number-one K3 surfaces of large
degree and remains uniform when their integral transcendental
Hodge-isometry class is prescribed.
\subsection{Notation and organization}
All varieties and function fields are over \(\C\), except for the
explicit generic-field arguments. Cohomology coefficients are
specified where used; transcendental lattices always mean integral
lattices modulo torsion. For reference, the main notation is collected below.
\begin{center}
\small
\begin{tabular}{@{}p{.18\textwidth}p{.72\textwidth}@{}}
\toprule
Symbol & Meaning \\
\midrule
\(T_M,\ q_M\) & Integral transcendental middle lattice of a fourfold and its cup form. \\
\(T_Z(-1)\) & Surface transcendental lattice shifted to weight four, with pairing \(-q_Z\). \\
\(G_M\) & Rational operator algebra on \(T_M\otimes\Q\), defined from genus-zero pairings in Definition~\ref{gw:algebra}. \\
\(\mathcal S,\ u\) & A degree-\(d\) K3 test collection and its MRC indicator. \\
\(c_u(f)\) & Target-acquired minus source-lost prime divisors, weighted by \(u\). \\
\(V/B,\ Q\) & A Mori fibre-space node and the common base of a link. \\
\(v,\ h_Q\) & Integer vertical-divisor correction and its portion dominating \(Q\); \eqref{sl:v-definition} and \eqref{sl:h-definition}. \\
\(w,\ \sigma_k,\ s\) & Integer corrections on a generic surface, a conic with a chosen surface subfield, and an entire Mori node; Definition~\ref{sl:w-def}, \eqref{sf:candidate}, and \eqref{sf:node-function}. \\
\bottomrule
\end{tabular}
\end{center}
Section~\ref{pre:section} defines the test and the divisorial cocycle.
Section~\ref{gw:section} proves the whole-lattice calculation in smooth
factorizations. Section~\ref{bd:section} establishes the bounded marked
diagrams and fixes the order of the uniform choices.
Section~\ref{sl:section} computes the correction on generic surfaces;
Section~\ref{sf:section} makes it intrinsic and proves vanishing by
telescoping. Section~\ref{cub:section} applies the resulting irrationality
criterion to every sufficiently large admissible Hassett divisor and
proves the three corollaries stated above.
==== END Irrational-cubic-fourfolds-with-geometric-K3-categories//introduction.tex ====
==== BEGIN Irrational-cubic-fourfolds-with-geometric-K3-categories//paper.tex ====
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\hypersetup{pdftitle={Irrational cubic fourfolds with geometric K3 categories},pdfsubject={Integral transcendental lattices, Gromov--Witten theory, and Sarkisov links},pdfauthor={OpenAI}}
\title{Irrational cubic fourfolds\\with geometric K3 categories}
\author{OpenAI}
\date{September 24, 2026}
\begin{document}
\maketitle
\begin{abstract}
For every sufficiently large admissible Hassett discriminant, we prove that
a very general cubic fourfold of that discriminant is irrational, although
its Kuznetsov component is equivalent to the ordinary derived category of a
projective K3 surface. The discriminant threshold is ineffective.
This disproves Kuznetsov's rationality conjecture.
The same cubics have associated untwisted polarized K3 surfaces in the
Hodge-theoretic sense, so they also disprove the sufficiency direction of
the associated-K3 rationality prediction.
\end{abstract}
\setcounter{tocdepth}{1}
\tableofcontents
\input{sections/introduction}
\input{sections/preliminaries}
\input{sections/gw}
\input{sections/bounds}
\input{sections/surface-links}
\input{sections/subfields}
\input{sections/cubics}
\bibliographystyle{plain}
\begingroup
\raggedright
\bibliography{references}
\endgroup
\end{document}
==== END Irrational-cubic-fourfolds-with-geometric-K3-categories//paper.tex ====
==== BEGIN Irrational-cubic-fourfolds-with-geometric-K3-categories//preliminaries.tex ====
\section{Birational fields and a divisorial cocycle}
\label{pre:section}
We define the field test and the additive birational invariant used in
both factorizations. On a smooth blowup the invariant will be the test
of the center, with a sign determined by the direction of the map.
All varieties and maps are over $\C$, unless a function field is explicitly
specified. A variety is integral. When a finite union is used, every sum is
taken over its integral components. A model of a finitely generated field
$F/\C$ is a variety with function field $F$; a smooth projective model may
always be chosen. A fourfold is \emph{rational} if it is birational over $\C$
to $\mathbb P^4$. Rational connectedness of a field means rational
connectedness of a smooth projective model.
We use characteristic-zero resolution and principalization in the
projective category, originating with Hironaka~\cite{Hironaka64}; precise
forms are given in \cite[Theorems~1.0.1--1.0.3]{Wlodarczyk05}.
Resolving the closure of a joint graph gives a common smooth projective
resolution of finitely many rational maps. Principalization makes the
marked ideals and exceptional boundary into a simple normal-crossing
support; see also \cite[Sections~1.2.2--1.2.3]{AKMW02}.
\subsection{The K3 test}
For a smooth projective variety $V$, write
$V\dashrightarrow R(V)$ for its maximal rationally connected quotient.
Its general fibres are rationally connected, its base is not uniruled,
and the base is determined up to birational equivalence. We use the
standard quotient theorem and the theorem that a fibration with
rationally connected base and general fibre has rationally connected
total space; see \cite[Chapter~IV, \S5]{Kollar96} and
\cite[Corollaries~1.3--1.4]{GHS03}.
The quotient has a proper morphism representative on suitable dense open
subsets \cite[\S0.7 and Theorem~2.3]{Campana92}.
Fix a positive integer $d>2$ and any collection $\mathcal S$ of
isomorphism classes of smooth projective K3 surfaces $S$ such that
\begin{equation}\label{pre:test-collection}
\operatorname{Pic}(S)=\Z L_S,\qquad L_S\text{ is ample},\qquad
L_S^2=d,\qquad \operatorname{Aut}_{\C}(S)=\{1\}.
\end{equation}
For odd $d$ the collection is empty. The collection can be restricted
by any further condition, including
an integral polarized transcendental Hodge-isometry condition. Every
degree threshold proved below is uniform in this choice of subcollection.
\begin{definition}[The test function]\label{pre:test}
For a finitely generated field $F/\C$, choose a smooth projective model
$V$ and set
\[
u(F)=
\begin{cases}
1,&\text{if }R(V)\text{ is birational to some }S\in\mathcal S,\\
0,&\text{otherwise}.
\end{cases}
\]
For an integral variety $V$, put $u(V)=u(\C(V))$.
\end{definition}
Birational invariance and uniqueness of the quotient make this definition
independent of the model. A K3 surface is not uniruled. Moreover, two
birational smooth projective K3 surfaces are isomorphic. One way to see
the latter assertion is to resolve a birational map as
$S\xleftarrow{p}W\xrightarrow{q}S'$. The pullbacks of the nowhere
vanishing two-forms on $S$ and $S'$ span the same one-dimensional space
$H^0(W,\Omega_W^2)$. Their zero divisors therefore have the same support.
These supports are respectively the exceptional loci of $p$ and $q$.
Every exceptional fibre is connected, so $q$ contracts every fibre of
$p$, and conversely. The image of $(p,q)$ in $S\times S'$ projects
properly, quasi-finitely and birationally to the normal surface $S$,
so that projection is an isomorphism.
Descending in both directions gives inverse morphisms. In particular,
$\operatorname{Bir}_{\C}(S)=\operatorname{Aut}_{\C}(S)$ for a K3 surface.
\begin{lemma}[Rationally connected fibrations]\label{pre:mrc-fibration}
Let $V\dashrightarrow W$ be a dominant rational map whose geometric
generic fibre is integral and has a rationally connected smooth proper
model. Then $R(V)$ and $R(W)$ are birational, and hence $u(V)=u(W)$.
In particular, for any vector bundle $\mathcal E$ of positive rank on
an integral variety $W$,
\[
u(\mathbb P_W(\mathcal E))=u(W).
\]
The test vanishes on rationally connected varieties and on varieties
of dimension at most one. For an integral surface $W$, it is one
precisely when $W$ is birational to a member of $\mathcal S$.
\end{lemma}
\begin{proof}
Resolve the rational maps and use smooth projective models.
Compose $V\dashrightarrow W$ with the maximal rationally connected
quotient of $W$. A general fibre of the composite fibres over a
rationally connected general fibre of $W\dashrightarrow R(W)$, with
rationally connected general fibre. It is rationally connected by the
fibration theorem of Graber--Harris--Starr
\cite[Corollary~1.3]{GHS03}. Since $R(W)$
is not uniruled, this composite is a maximal rationally connected
quotient of $V$. Indeed, a family of rational curves through general
points of $V$ that is not vertical would induce a covering family of
rational curves on $R(W)$. Uniqueness of the quotient proves the first
assertion. A projective bundle has a projective-space generic fibre.
The remaining statements follow from the dimension of the quotient,
and from the fact that a non-uniruled surface is its own quotient.
\end{proof}
We also use $u$ on finite field extensions and finite Galois sets.
If $k/\C$ is finitely generated and a finite \mbox{\'{e}tale}
$k$-algebra is $A=\prod_i k_i$, define
\begin{equation}\label{pre:finite-orbits}
u(A)=\sum_i u(k_i).
\end{equation}
Equivalently, for a finite set with continuous action of
$\operatorname{Gal}(\bar k/k)$, sum once over its Galois orbits, using
the finite residue field belonging to each orbit. There is no factor
$[k_i:k]$ in this sum. Each $k_i$ is viewed as a field over $\C$.
Geometric generic fibres over algebraic closures of function fields will
be used to obtain bounds, while $u$ continues to be evaluated on the
original finitely generated fields over $\C$.
\begin{lemma}[Small orbits]\label{pre:small-orbits}
Suppose $k\simeq\C(x,y)$. A finite Galois orbit of size one or two
has test value zero. More generally, if a finite extension $\ell/k$
has a nonidentity $\C$-automorphism, then $u(\ell)=0$.
\end{lemma}
\begin{proof}
The field $k$ is rational. If $u(\ell)=1$, the surface field $\ell$
is the function field of a member of $\mathcal S$, by
Lemma~\ref{pre:mrc-fibration}. A nonidentity field automorphism would
give a nonidentity birational, hence regular, automorphism of that K3
surface. This contradicts the choice of $\mathcal S$. A quadratic
extension in characteristic zero has its nonidentity involution.
\end{proof}
\subsection{Prime divisors as valuations}
Let $X$ be a normal projective variety with function field $F$.
A prime divisor $E\subset X$ determines the normalized discrete
valuation $\operatorname{ord}_E:F^*\to\Z$, whose residue field is
$\C(E)$. Write $\mathcal D_F(X)$ for this set of valuations, using
a specified identification $\C(X)\simeq F$ when necessary. Distinct
valuations are counted separately even when their residue fields are
isomorphic.
For a birational map $f:X\dashrightarrow Y$, identify the function
fields by $f^*: \C(Y)\xrightarrow{\sim}\C(X)=F$ and define
\begin{equation}\label{pre:cocycle-definition}
c_u(f)=
\sum_{\nu\in\mathcal D_F(Y)\setminus\mathcal D_F(X)}u(\kappa(\nu))
-\sum_{\nu\in\mathcal D_F(X)\setminus\mathcal D_F(Y)}u(\kappa(\nu)).
\end{equation}
Thus divisors acquired by the target have positive sign. The target's
valuation set is transported using $f^*$ even when $X=Y$.
This is the divisorial invariant of Lin and Shinder, composed with the
MRC/K3 test \(u\); the sign agrees with their target-acquired minus
source-lost convention
\cite[Definition~2.1, Lemma~2.2 and Proposition~2.3]{LinShinder24}.
\begin{lemma}[Divisorial cocycle]\label{pre:cocycle}
The sums in \eqref{pre:cocycle-definition} are finite. For birational
maps $f:X\dashrightarrow Y$ and $g:Y\dashrightarrow Z$ of normal
projective varieties,
\begin{equation}\label{pre:cocycle-additivity}
c_u(g\circ f)=c_u(f)+c_u(g),\qquad
c_u(f^{-1})=-c_u(f).
\end{equation}
An isomorphism in codimension one has value zero.
If $\pi:\widetilde X\to X$ is the blow-up of a smooth projective
variety along a smooth center with connected components $Z_i$ of
codimension at least two, then
\begin{equation}\label{pre:blowup-sign}
c_u(\pi^{-1})=\sum_i u(Z_i),\qquad
c_u(\pi)=-\sum_i u(Z_i).
\end{equation}
\end{lemma}
\begin{proof}
Choose dense open subsets on which $f$ is an isomorphism. Every divisor
whose generic point belongs to these opens has a corresponding divisor
on the other model with exactly the same valuation. Any valuation in
the symmetric difference must therefore be represented by a divisorial
component of one of the two closed complements. There are finitely many
such components. This also proves that an isomorphism in codimension
one has value zero.
For the composition use the common field $F=\C(X)$, transporting
$\C(Y)$ by $f^*$ and $\C(Z)$ by $f^*g^*$. Put
$D_X=\mathcal D_F(X)$, $D_Y=\mathcal D_F(Y)$ and
$D_Z=\mathcal D_F(Z)$. Their pairwise symmetric differences are finite.
On the set of all prime divisorial valuations of $F$,
\[
\mathbf 1_{D_Z}-\mathbf 1_{D_X}
= (\mathbf 1_{D_Y}-\mathbf 1_{D_X})
+(\mathbf 1_{D_Z}-\mathbf 1_{D_Y}).
\]
All three differences have finite support. Multiplying by
$u(\kappa(\nu))$ and summing proves additivity. Transport by a
$\C$-field isomorphism identifies residue fields, so the last term
is precisely $c_u(g)$. Reversing the two valuation sets gives the
inverse formula.
For the blow-up, the target of the extraction
$\pi^{-1}:X\dashrightarrow\widetilde X$ retains every original
divisorial valuation and adds exactly one exceptional divisor
$E_i=\mathbb P_{Z_i}(N_{Z_i/X})$ for each $Z_i$. Its residue field is
purely transcendental over $\C(Z_i)$, of transcendence degree
$\operatorname{codim}_X(Z_i)-1$. Lemma~\ref{pre:mrc-fibration} gives
$u(E_i)=u(Z_i)$. The stated signs now follow from the definition and
the inverse formula.
\end{proof}
For the generic-surface interpretation, let $V\to Q$ have a smooth
surface generic fibre over $k=\C(Q)$. Blowing up a closed point $P$ of
that fibre creates the exceptional curve $\mathbb P^1_{\kappa(P)}$.
Its function field is $\kappa(P)(t)$, so the corresponding divisorial
valuation of $\C(V)$ contributes $u(\kappa(P))$ once, by
Lemma~\ref{pre:mrc-fibration}. There is no factor $[\kappa(P):k]$.
This is the compatibility between \eqref{pre:finite-orbits} and the
divisorial count used for generic surface links.
\subsection{The smooth factorization used below}
\begin{theorem}[Weak factorization]\label{pre:weak-factorization}
A birational map between smooth projective varieties over $\C$ factors
into a finite zigzag of blow-ups and blow-downs along smooth connected
centers, with all intermediate varieties smooth and projective.
Nontrivial steps may be taken to have centers of codimension at least two.
\end{theorem}
This is the projective case of
\cite[Theorem~0.1.1]{AKMW02}; blowing up a smooth divisor is an
isomorphism and may be omitted. In a factorization of a birational map
from $\mathbb P^4$, every intermediate variety is itself a smooth
projective rational fourfold, hence rationally connected. Its centers
have dimension zero, one, or two. Lemma~\ref{pre:cocycle} computes
$c_u$ as the signed sum of their tests along the zigzag; only surface
centers can have nonzero test.
==== END Irrational-cubic-fourfolds-with-geometric-K3-categories//preliminaries.tex ====
==== BEGIN Irrational-cubic-fourfolds-with-geometric-K3-categories//subfields.tex ====
\section{An intrinsic correction on conic-bundle nodes}\label{sf:section}
The surface calculation depends on a surface field inside the field of a
three-dimensional conic-bundle base. We now remove that dependence. All
fields and their inclusions in this section are over $\C$. A function field
attached to a model is identified with its specified subfield of the common
function field; replacing the model does not replace that inclusion.
The test $u$ is the test of Section~\ref{pre:section}, for a collection of
Picard-number-one K3 surfaces of degree $d$ having no nonidentity
automorphism. None of the bounds below depends on which subcollection is
used. We will show that a nonzero correction singles out its embedded
surface field, and that this correction vanishes at the ends of links
over a point or curve. These two assertions will make the surface
calculation telescope along a fourfold factorization.
We retain the convention of Section~\ref{sl:section} that ordinary
four-dimensional links belong to the paths chosen in
Theorem~\ref{bd:diagrams}\textup{(i)}.
\subsection{Eligible surface fields and the candidate}
Let $L$ be the function field of the three-dimensional base of a conic
Mori fibre space, and let $\alpha\in\operatorname{Br}(L)[2]$ be the
class of its generic conic. The zero class is allowed. A subfield
$k\subset L$ is \emph{eligible} if it is a surface function field and
$L$ is the function field of a smooth projective geometrically integral
genus-zero curve $C_k$ over $k$, with the following ramification bound.
Let $\Delta_k\subset C_k$ be the reduced finite subscheme of closed points
at which $\alpha$ has nonzero residue. We require
\begin{equation}\label{sf:eligibility}
r_k:=\deg_k\Delta_k\leq 7.
\end{equation}
Thus $r_k$ counts geometric points of the ramification support defined
\emph{before} extending the constants. It does not mean the ramification
of the restriction of $\alpha$ after extension to an algebraic closure
of $k$.
The residue at each point of $\Delta_k$ defines a quadratic extension of
its residue field. Together these give a degree-two finite \'etale cover
$\widetilde\Delta_k\to\Delta_k$. If $r_k=3$, let $P_k$ be the finite
\'etale $k$-scheme whose geometric points choose one point in each of the
three fibres of this double cover. It has degree eight. Set
\begin{equation}\label{sf:candidate}
\sigma_k(L,\alpha)
=u(\Delta_k)+\mathbf{1}_{r_k=3}\,\frac12u(P_k).
\end{equation}
Here $u$ on a finite \'etale scheme is the sum of $u$ on its field factors,
as in the surface calculation. The data in
\eqref{sf:candidate} are intrinsic to $(k\subset L,\alpha)$: a genus-zero
function field has a unique smooth projective curve model, and Brauer
residues are defined at its discrete valuations.
For the presentation supplied by a nontrivial generic two-ray move,
Proposition~\ref{sl:surface-identity} gives $r_k\leq7$.
Lemma~\ref{sl:conic-h} and Definition~\ref{sl:w-def} then identify
\eqref{sf:candidate} with the local correction $w-h_Q$.
The present definition also allows other eligible embedded fields;
we must compare their candidates before assigning a value to the node.
The half in \eqref{sf:candidate} causes no integrality ambiguity.
Complementing all three choices defines a fixed-point-free involution of
the geometric set $P_k$. A field factor preserved by this involution
has a nonidentity automorphism. If it were counted by $u$, this would
induce a nonidentity birational automorphism of its K3 minimal model;
birational maps between smooth minimal K3 surfaces are isomorphisms.
Such a factor is therefore not counted. The counted factors occur in
pairs interchanged by complementation. In particular $u(P_k)$ is even.
This observation is not needed for telescoping, but shows that all the
candidate corrections are nonnegative integers.
\subsection{Bounded ramification diagrams}
The following lemma supplies the implication needed in both comparison
arguments: a bounded diagram with marked ramification forces the
candidate to vanish at large degree. We will then construct log paths
whose markings satisfy its hypothesis. Boundedness includes the
morphisms and embedded marked support, rather than only the isomorphism
classes of their sources and targets, as in
Theorem~\ref{bd:diagrams}.
\begin{lemma}[Residues in a bounded diagram]\label{sf:residue-bound}
Suppose that a normal projective model of a genus-zero presentation
$L/k$ occurs in
a bounded family of diagrams
\[
Y\longrightarrow S,
\]
where $S$ is a normal projective surface model of $k$. Suppose also
that the family has
bounded marked divisorial support containing the full visible
ramification of $\alpha$ on $Y$. The family may instead be given on
the geometric generic fibre over a curve $Q$, in which case the
diagrams are over $Q$ and $\C(Q)\subset k$ is required.
For $r_k\leq7$, the orbit fields in $\Delta_k$, $\widetilde\Delta_k$,
and, when defined, $P_k$, are finite covers of degree at most fourteen
with branch contained in a uniformly bounded bad locus on the base
diagram. Consequently none of these orbit fields is counted by $u$
once $d$ exceeds a constant depending only on the bounded family.
\end{lemma}
\begin{proof}
Resolve the diagram and marked support in bounded families, marking all
resolution exceptional divisors as well. Write $D$ for the resulting
SNC support on the smooth total space. It is enough that $D$ contain
the ramification; its unramified components do no harm. Every prime
on the resolution is either the strict transform of a prime on the
old model or an exceptional prime. Marking all exceptional primes
therefore includes any newly visible ramification, even over the
singular locus of the old model.
We form residues and their quadratic covers before extending constants:
the ground field $F$ is $\C$, or $\C(Q)$ on the ordinary generic fibre
over $Q$. After finding their unramified open, we will base change
these same covers to $\overline{\C(Q)}$ to use the geometric bounds.
The Kummer
isomorphism identifies the Brauer $2$-torsion of the total
function field with its $H^2$ with coefficients in $\mu_2$
\cite[\S2, equations~(1)--(4)]{AuelBohningBothmerPirutka20}. On the
smooth total model, the first two residue maps form a complex
\[
H^2(F(Y),\mu_2)
\xrightarrow{\partial}
\bigoplus_{E\in Y^{(1)}}H^1(F(E),\Z/2)
\xrightarrow{\partial}
\bigoplus_{z\in Y^{(2)}}H^0(F(z),\mu_2^{-1}),
\qquad \partial^2=0.
\]
Here $Y^{(j)}$ denotes the codimension-$j$ points. These are the
coniveau differentials of
\cite[Example~2.1 and Proposition~3.9]{BlochOgus74}; the same complex
applies over the characteristic-zero function field of $Q$.
Consider a ramification component $D_i$ and a codimension-one point
$z$ of $D_i$ outside its intersections with the other components of
$D$. Every term except that from $D_i$ is zero in the displayed
secondary residue at $z$. Since $D_i$ is smooth, that term is its
ordinary DVR residue, so $\partial_z(\partial_{D_i}\alpha)=0$.
If the quadratic residue is represented by
$a\in F(D_i)^*/F(D_i)^{*2}$, this says that
$\operatorname{ord}_z(a)$ is even. After multiplication by a square,
$a$ is a unit at the DVR, and adjoining its square root is unramified
there. Normalize the indicated open of $D_i$ in the quadratic
algebra, componentwise in the split case. This normalization is
finite, is unramified at every codimension-one point, and has regular
target. Purity of the branch locus makes it \'etale throughout that
open \cite[Lemma~58.21.4, Tag~0BMB]{StacksPurityBranch2026}.
The horizontal ramification components are generically finite over
$S$. Delete their nonfinite and branch loci on $S$, the images of their
pairwise intersections, the images of their intersections with the
remaining marked support, and the singular and resolution bad loci of
the diagram. These images are proper closed subsets: over the generic
point of $S$ the distinct horizontal components are distinct points
of the smooth proper curve. The deleted sets form a bounded embedded
boundary after stratifying the family. Over the resulting smooth open
$U\subset S$, the horizontal union is finite \'etale, its components are
smooth and disjoint from the other marked divisors, and its residue
double cover is finite \'etale. Their degrees over $U$ are $r_k$ and
$2r_k$. The scheme of sections defining $P_k$ is then a finite \'etale
scheme of degree eight over $U$. Thus all the fields in the statement
have the asserted degree and branch bounds.
These degree and branch bounds exclude the large-degree K3 fields.
First consider a bounded diagram over $\C$.
Fix an orbit field $E/k$ in the statement, and let $S_E\to S$ be
the normalization in $E$. Take a general pencil in a uniformly bounded
very ample system on the resolved base surface $S$. Let $h$ bound the
genus of its normalized general member and let $b$ bound its intersections
with the deleted boundary. Each normalized component of the pullback
of that member to $S_E$ has degree $n\leq14$ and, by Hurwitz
\cite[Lemmas~53.12.2 and~53.12.4]{StacksRiemannHurwitz2026},
\begin{equation}\label{sf:hurwitz}
2g-2\leq n(2h-2)+b(n-1).
\end{equation}
If $u(E)=1$, a smooth projective model of $S_E$ is birational to a
test K3 surface. Resolve the pulled-back pencil on this model and take
its Stein factorization. Its connected general fibre is one of these
normalized components. This contradicts the bound
$2g-2\geq\sqrt d$ in Lemma~\ref{bd:pencil} for large $d$.
When the family is given only on the geometric generic fibre of
$Q$, the same construction is performed over
$\overline{\C(Q)}$ on the base curve there. Finite \'etale covers remain
finite \'etale after this extension, although they may split. The same
degree and Hurwitz bounds therefore hold on every geometric component.
We retain the base changes of the original ramification support and
residue covers; we do not recompute the support from the extended
Brauer class, whose residues may have become zero.
For any orbit field $E/k$, the inclusions $\C(Q)\subset k\subset E$
give a rational map from its surface model to $Q$.
If $u(E)=1$, resolve that map on the corresponding K3 surface and take
its Stein factorization. Then
the geometric general fibres of that map are precisely the components
whose genera were bounded. Lemma~\ref{bd:pencil} again gives the
contradiction. No assertion about the complexity of an unmarked
Brauer class is used.
\end{proof}
\subsection{The coefficient and the full visible boundary}
We use the single rational coefficient supplied by
Corollary~\ref{bd:choice-order}. If $c_*>0$ denotes the uniform upper
bound obtained there from the ordinary conic-base diagrams, this choice is
\begin{equation}\label{sf:c-choice}
0<c<\min\{1/4,c_*\}.
\end{equation}
With $c$ fixed, take the log three-dimensional diagram bounds of
Theorem~\ref{bd:diagrams} and the terminalization bounds of
Theorem~\ref{bd:terminalization}. Only then fix the common lower bound
on $d$ required by Lemma~\ref{sf:residue-bound} and
Theorem~\ref{bd:exclusions} for these families. These choices are
uniform in the eligible fields, test subcollections, factorizations,
and their lengths.
Here are the boundary conventions needed to apply those results.
Given finitely many presentations of $L$, take a common smooth
projective model $Y$ resolving the rational maps to projective models
of their bases. Resolve the ramification support and let $D_Y$ be
\emph{exactly} the reduced divisor of ramification visible on this
model. Use the pair
\begin{equation}\label{sf:start}
(Y,\Phi),\qquad \Phi=cD_Y.
\end{equation}
The divisor $D_Y$ has SNC support. All models and maps in the log
geography are taken with the transformed support; they do not extract
prime valuations from $Y$. Their transformed support is therefore
the full ramification visible on them. If a higher common start is
needed, use its full visible ramification again, including the
ramified exceptional divisors that have appeared.
This last convention is compatible with the discrepancy requirement
in Theorem~\ref{bd:diagrams}. For the first blowup of a smooth centre
of codimension $\ell\geq2$ contained in $r\leq\ell$ boundary components,
the log discrepancy is $a(F,Y,cD_Y)=\ell-cr$. Its ordinary discrepancy
is therefore $a(F,Y,cD_Y)-1=\ell-1-cr\geq1-2c>0$.
For an arbitrary exceptional prime, the SNC inequality
\eqref{bd:snc-discrepancy} gives $a(F,Y,cD_Y)\geq2(1-c)>1$.
On a higher smooth start $\pi:Y'\to Y$, the coefficient of $F$ in
$K_{Y'}+cD_{Y'}-\pi^*(K_Y+cD_Y)$ is
$a(F,Y,cD_Y)-1+c\epsilon_F$, where $\epsilon_F$ is one if $F$ is
ramified and zero otherwise. This coefficient is positive in either
case. A negative log MMP increases discrepancies for contracted primes.
Lemma~\ref{bd:higher-start} therefore recovers the same endpoints, and
the common-start clauses of Theorem~\ref{bd:diagrams} apply with the
fixed coefficient in \eqref{sf:c-choice}.
For an eligible $k$, the degree on the generic curve is
\begin{equation}\label{sf:negative-degree}
\deg(K+\Phi)=-2+cr_k\leq-2+7c<-\tfrac14<0.
\end{equation}
Run the log MMP over a surface model of $k$. The relative programme
ends in a Mori fibre space over a two-dimensional base birational to
that surface; its embedded base field is still $k$. Indeed the
generic relative dimension is one, and the final generic contraction
is a genus-zero curve to its ground field. Geometric integrality
makes this ground field $k$ rather than a proper finite extension.
These relative negative steps are also permissible steps in the
common-start log geography over projective bases.
\subsection{Uniqueness of the eligible field}
\begin{proposition}[Uniqueness from a nonzero candidate]
\label{sf:unique}
For $d$ above the uniform bound fixed after \eqref{sf:c-choice}, if
an eligible $k\subset L$ has
$\sigma_k(L,\alpha)\ne0$, then every eligible surface subfield of
$(L,\alpha)$ equals $k$ as an embedded subfield of $L$.
\end{proposition}
\begin{proof}
Let $k'\subset L$ be any other eligible field; its candidate need not
be nonzero. Resolve the two presentations on a common start
\eqref{sf:start}. By \eqref{sf:negative-degree}, its log MMP has an
output with base field $k$ and another with base field $k'$. Connect
these outputs by the log Sarkisov links of
Theorem~\ref{bd:diagrams}, and write their successive nodes as
$Z_i/S_i$ for $0\leq i\leq N$. The birational maps from the common
start identify every $\C(Z_i)$ with $L$; set $K_i=\C(S_i)\subset L$.
Thus $K_0=k$ and $K_N=k'$. The class $\alpha\in\operatorname{Br}(L)[2]$
is fixed throughout this path, and the support of each boundary is its
full visible ramification on the corresponding model.
Every Mori base in this three-dimensional programme has dimension at
most two. If a common base of a link has dimension two, the endpoint
bases also have dimension two and their contractions to the common
base are birational: they are connected-fibre contractions between
normal varieties of the same dimension. The induced identifications
are the ones coming from $L$. Such a link therefore preserves the
embedded surface field.
If $k'\ne k$, let $m$ be the first index for which $S_{m+1}$ is not
a surface with $K_{m+1}=k$. Its common base $Q$ has dimension at most
one, by the preceding paragraph. At each preceding node the data
$(k\subset L,\alpha)$ are unchanged. Hence its proper generic curve,
residue covers, and candidate are the original $C_k$, the original
covers, and $\sigma_k(L,\alpha)$. The relevant part of the path is
\[
\begin{tikzcd}[column sep=large,row sep=large]
Z_m \arrow[rr,dashed,"\text{first departure}"] \arrow[d]
&& Z_{m+1}\arrow[d] \\
S_m\arrow[dr] && S_{m+1}\arrow[dl] \\
& Q
\end{tikzcd}
\qquad
\begin{gathered}
\C(S_0)=\cdots=\C(S_m)=k\subset L,\\
\dim Q\leq1.
\end{gathered}
\]
By Theorem~\ref{bd:diagrams}, on the
geometric generic fibre of $Q$ its whole marked diagram is uniformly
bounded, including the projection to the endpoint surface base.
All ramification points on the proper generic curve are visible as
divisors on this total model: their closures dominate the surface
base and have codimension one. They are among the marked transforms
by the full visible convention. If $Q$ is a curve, its function
field is contained in $k$ because the endpoint base contracts to
$Q$. Lemma~\ref{sf:residue-bound} applies and gives
$\sigma_k(L,\alpha)=0$, a contradiction. Thus $k'=k$.
\end{proof}
\subsection{Vanishing at ends over smaller common bases}
At ends of ordinary four-dimensional links over a point or curve,
we must exclude both the del Pezzo correction on a surface base and
the eligible-field candidates on a three-dimensional conic base.
The surface-base case uses the bounded finite sets already defined;
the conic case will require a second log MMP output.
\begin{lemma}[Vanishing at surface bases]\label{sf:surface-end}
At an end $V\to B$ with $\dim B=2$ of an ordinary four-dimensional
link over a common base $Q$ of dimension at most one, the
degree-five or degree-six del Pezzo correction $w$ is zero for
uniformly large $d$.
\end{lemma}
\begin{proof}
The relative generic diagram over $Q$ is bounded by
Theorem~\ref{bd:diagrams}. On the smooth del Pezzo locus, let
$a\in\{5,6\}$ be the fibre degree. The vanishings
$H^1(\mathcal O)=H^2(\mathcal O)=0$ make the relative Picard scheme
\'etale
\cite[Theorem~4.8, Corollary~5.13 and Proposition~5.19]{KleimanPicard057}.
We identify its finite part that parametrizes conic-pencil classes.
For a line-bundle class $F$ on a geometric fibre $S$, Riemann--Roch gives
\[
\chi\bigl(\mathcal O_S(F-mK_S)\bigr)
=1+\tfrac12\bigl(F^2-(2m+1)F\cdot K_S+a m(m+1)\bigr).
\]
Consequently the equations $F^2=0$ and $-K_S\cdot F=2$ are equivalent
to having the fixed anticanonical Hilbert polynomial
$\tfrac a2m(m+1)+2m+2$.
This fixed-polynomial locus in the relative Picard scheme is open and
closed and of finite type; in the smooth projective family it is also
projective
\cite[Theorem~6.20, Exercise~5.7 and its answer]{KleimanPicard057}.
It is therefore proper and quasi-finite, hence finite \'etale.
Lemma~\ref{sl:pencil-sets} identifies its geometric fibres
with $A(S)$, of cardinality five or three according as $a=5$ or $6$.
These are statements about the Picard scheme and require no universal
line bundle on the original family.
The complement of this smooth family is bounded as an embedded locus
of the diagram. Each field factor of the finite scheme is
therefore a cover of bounded degree unramified outside this bounded
locus. Apply the bounded-cover exclusion of
Theorem~\ref{bd:exclusions}, or the Hurwitz calculation
\eqref{sf:hurwitz}. When $Q$ is a curve its field is contained in
each tested surface field. Thus no factor is counted by $u$ and
$w=0$.
\end{proof}
Now let $V\to T$ be a conic end with $\dim T=3$ and
common base $Q$ of dimension at most one, and put $F=\C(Q)$.
Corollary~\ref{bd:choice-order} gives a bounded family of klt log-Fano
pairs $(T_{\overline F},c(D_T)_{\overline F})$ with the retained marked
support and the fixed coefficient \eqref{sf:c-choice}. These are the
inputs to the application of Theorem~\ref{bd:terminalization} in the
following proof. The corollary proves
rank one over $F$; it requires no rank-one assertion over $\overline F$.
\begin{proposition}[Vanishing at conic ends]\label{sf:conic-end}
Let $V\to T$ be a conic end of an ordinary four-dimensional Sarkisov
link with common base $Q$ of dimension at most one. For the uniform
large degrees under consideration, no eligible surface subfield of
$(\C(T),\alpha)$ has nonzero candidate.
\end{proposition}
\begin{proof}
Put $L=\C(T)$ and suppose an eligible $k\subset L$ has nonzero
candidate. The bounded diagram on $T$ does not yet include the
presentation over $k$; in particular, $k$ need not contain $\C(Q)$.
We will construct an output with bounded marked geometric generic
diagram over $Q$ and compare it with the output over $k$.
Choose a common smooth
SNC start $Y$ mapping to $T$ and to a projective surface model of
$k$, with the full visible boundary \eqref{sf:start}. One log MMP
output has base field $k$ by \eqref{sf:negative-degree}.
\paragraph{A bounded output over $Q$.}
First run the log MMP \emph{over $T$} to a relative minimal model
$b:Y^*\to T$. This is the birational case: relative bigness holds,
so \cite[Theorem~1.2]{BCHM10} applies. Let $D_*$ be the transform
of $D_Y$ on $Y^*$; its pushforward under $b$ is $D_T$.
We first study the geometric generic fibre over $Q$, retaining the
same symbols for these restrictions. The marked supports
are the base changes of the original supports, regardless of whether
the extended Brauer class still ramifies on them. We have
\begin{equation}\label{sf:exceptional-nef}
K_{Y^*}+cD_* = b^*(K_T+cD_T)+E.
\end{equation}
The divisor $E$ is $b$-exceptional and $b$-nef; the latter follows
from relative nefness of the left side. The negativity lemma gives
$E\leq0$. For an exceptional prime $F$ of $b$, its coefficient is
\begin{equation}\label{sf:cutoff}
\operatorname{coeff}_F(E)
= a(F,T,cD_T)-1+c\epsilon_F,
\qquad
\epsilon_F=\begin{cases}1&F\text{ occurs in }D_*,\\0&F\text{ does not occur in }D_*.
\end{cases}
\end{equation}
Consequently every valuation extracted by $b$ on this geometric
generic fibre satisfies
$a(F,T,cD_T)\leq1$; an extracted valuation occurring in $D_*$ even satisfies
$a(F,T,cD_T)\leq1-c$.
Theorem~\ref{bd:terminalization} now bounds the geometric generic
model and all its subsequent non-extracting contraction diagrams,
including the marked support. Indeed the only extracted divisors
are among the bounded terminalization data for the bounded klt log
Fano pairs $(T,cD_T)$. On the geometric generic fibre over $Q$,
every component of $D_*$ is either the strict transform of a component
of $D_T$, or one of these extracted prime divisors. The markings here
are retained under extension of constants. The terminalization
theorem marks the entire extraction list, including primes with
crepant coefficient zero, so it bounds this full support and all of
its subsequent transforms on that geometric generic fibre. Primes
of the resolution exceptional over $T$ there and absent from this
list cannot survive on $Y^*$ there by \eqref{sf:cutoff}.
This bounds the geometric generic model $Y^*$ over $Q$ and its
subsequent non-extracting diagrams, not the arbitrary resolution $Y$.
To obtain a Mori output over $Q$, we verify
that the log canonical divisor on this geometric generic fibre is not
pseudo-effective. To see this from \eqref{sf:exceptional-nef}, choose
a general complete-intersection curve in $T$ avoiding the centres
of the exceptional divisors of $b$. Its strict transform has
negative intersection with $K_{Y^*}+cD_*$, because $K_T+cD_T$ is
anti-ample. Such curves form a movable covering family, so a
pseudo-effective divisor could not have this negative intersection.
Return to the models over $\C$. We may therefore run the log MMP
from $Y^*$ over $Q$ to a Mori fibre space $Z/B$. Its geometric
generic diagram over $Q$ is bounded by the terminalization theorem,
since these additional steps do not extract divisors.
\paragraph{Comparison with the output over $k$.}
Connect the output with field $k$ to $Z/B$ by log links from the same
initial pair. These links are taken in the absolute common-start
geography; the arbitrary eligible field $k$ need not contain $\C(Q)$.
If $B$ is not a surface with embedded field $k$, consider the first
departure from $k$ along this path, as in Proposition~\ref{sf:unique}.
Call the common base of that link $Q'$. It has dimension at most one,
its incoming endpoint has the original candidate $\sigma_k(L,\alpha)$,
and, when $Q'$ is a curve, its morphism to $Q'$ gives
$\C(Q')\subset k$. The bounded diagram over $Q'$ therefore forces
that candidate to vanish by Lemma~\ref{sf:residue-bound}.
If instead $B$ is a surface with $\C(B)=k$, the bounded geometric
generic diagram of $Z/B$
over the original $Q$ gives the same contradiction directly. Here
$B\to Q$ gives $\C(Q)\subset k$ when $Q$ is a curve, so the other
application of Lemma~\ref{sf:residue-bound} has its required field
inclusion as well. Both alternatives contradict the nonzero candidate.
\end{proof}
\subsection{The intrinsic function and telescoping}
We now attach the correction to a four-dimensional Mori node $V\to B$.
When $\dim B=3$, let
$\alpha\in\operatorname{Br}(\C(B))[2]$ be the class of the generic
conic. Define a function on the entire node by
\begin{equation}\label{sf:node-function}
s(V/B)=
\begin{cases}
0,&\dim B\leq1,\\
w(V/B),&\dim B=2,\\
\sigma_k(\C(B),\alpha),&\dim B=3\text{ and some eligible }k
\text{ has nonzero candidate},\\
0,&\dim B=3\text{ and no such }k\text{ exists}.
\end{cases}
\end{equation}
The function $w$ in base dimension two is the del Pezzo function
of Definition~\ref{sl:w-def}.
\begin{theorem}[Intrinsic correction]\label{sf:intrinsic}
For uniformly sufficiently large $d$, the function
\eqref{sf:node-function} is well defined. At a conic node its value
equals the candidate of \emph{every} eligible embedded surface
field, including fields whose candidate is zero. At either end
of an ordinary four-dimensional link with common base of dimension
at most one, its value is zero.
\end{theorem}
\begin{proof}
If there is a nonzero candidate, Proposition~\ref{sf:unique} says
that its eligible field is the only eligible field. If there is
no nonzero candidate, all eligible candidates are zero and agree
with the definition. In particular a zero candidate cannot be
ignored in favour of a nonzero candidate from another field.
The last assertion follows from Proposition~\ref{sf:conic-end},
Lemma~\ref{sf:surface-end}, and the definition when $\dim B\leq1$.
\end{proof}
\begin{theorem}[Vanishing between Mori spaces over points]
\label{sf:vanishing}
There is a bound $d_0$ with the following property. Let $X_1$ and
$X_2$ be normal projective $\Q$-factorial terminal rationally connected
complex fourfolds whose maps to $\operatorname{Spec}\C$ are Mori
fibre spaces. For $d>d_0$, every birational map
$f:X_1\dashrightarrow X_2$ satisfies
\[
c_u(f)=0.
\]
The bound is independent of $X_1$, $X_2$, $f$, its Sarkisov
factorization, and the chosen degree-$d$ test subcollection.
\end{theorem}
\begin{proof}
Choose an ordinary Sarkisov factorization supplied by
Theorem~\ref{bd:diagrams}, and orient each link from
$V/B$ to $V'/B'$. Write $\Delta$ for final minus initial value of
a function on its nodes. Proposition~\ref{sl:vertical} gives
\begin{equation}\label{sf:residual}
c_u(\text{link})-\Delta v
=H_Q=c_{u,\mathrm{hor}/Q}-\Delta h_Q.
\end{equation}
We check $H_Q=\Delta s$ in every dimension of the common base $Q$.
If $\dim Q\leq1$, Proposition~\ref{sl:low-base} gives $H_Q=0$, while
Theorem~\ref{sf:intrinsic} gives $s=0$ at both ends.
If $\dim Q=3$, both endpoint bases have dimension three and are
birational to $Q$. On the generic fibre the link is an isomorphism
of smooth projective conics. Thus the endpoint base fields and
quaternion classes are identified, including all eligible embedded
surface fields and their candidates. Therefore $\Delta s=0$,
which equals $H_Q$ by Proposition~\ref{sl:low-base}.
Suppose $\dim Q=2$ and put $k=\C(Q)$. In case~\textup{(ii)} of
Lemma~\ref{sl:generic-dichotomy}, the calculation of
Proposition~\ref{sl:surface-identity} gives
$c_{u,\mathrm{hor}/Q}=\Delta w$. Here $w$ is evaluated on the
structured generic surface over $k$, including its conic structure
when $\dim B=3$, as in Definition~\ref{sl:w-def}.
On a del Pezzo side,
$h_Q=0$ and $s=w$. On a conic side, its generic curve base over
$k$ is geometrically integral of genus zero. Lemma~\ref{sl:conic-h}
identifies its singular-fibre set and component cover with
$\Delta_k$ and the residue cover. The common del Pezzo surface $W$
has positive degree, so \eqref{sl:conic-size} gives
$r_k=\deg_k\Delta_k=8-K_W^2\leq7$. Hence $k$ is eligible.
The same lemma gives $h_Q=-u(\Delta_k)$, and by
\eqref{sf:candidate} and Theorem~\ref{sf:intrinsic},
\[
s=w-h_Q.
\]
This remains valid if the candidate is zero. Consequently
$H_Q=\Delta(w-h_Q)=\Delta s$.
In case~\textup{(i)} of Lemma~\ref{sl:generic-dichotomy}, the
structured generic surfaces are isomorphic after cancelling any
identical generic contractions. Their exceptional horizontal
contribution and $H_Q$ are zero by that lemma. At a
del Pezzo node the Picard-class sets defining $w$ identify. At a
conic node the isomorphism identifies the embedded base field
and its quaternion class, so the intrinsic corrections identify.
Thus $\Delta s=0$ here as well. This last argument requires no
bound on the discriminant degree of an isomorphic conic
structure.
It follows from \eqref{sf:residual} that every link satisfies
\[
c_u(\text{link})=\Delta(v+s).
\]
Add over the factorization and use the cocycle property
of Lemma~\ref{pre:cocycle}. At a node over a point there
are no vertical divisors in the defining difference for $v$,
and $s=0$ by definition. The sum therefore vanishes. All lower
bounds on $d$ arose from the finitely many fixed families specified
after \eqref{sf:c-choice}; none depends on the length or starting
resolution of the factorization.
\end{proof}
==== END Irrational-cubic-fourfolds-with-geometric-K3-categories//subfields.tex ====
==== BEGIN Irrational-cubic-fourfolds-with-geometric-K3-categories//surface-links.tex ====
\section{Divisorial corrections and surface links}
\label{sl:section}
We first remove the contributions vertical over the common base of a link.
When that base has dimension two, the remaining surface calculation
produces a correction depending on its function field;
Section~\ref{sf:section} will make the correction intrinsic to each
Mori fibre space for uniformly large test degree.
\subsection{Vertical divisors and change of base}
Fix a degree $d$ and a subcollection $\mathcal S$ of the Picard-number-one
K3 surfaces of degree $d$ with trivial automorphism group. Write $u(Y)$ for
the indicator that the maximal rationally connected quotient of a smooth
proper model of $\C(Y)$ is birational to a member of $\mathcal S$.
As in Definition~\ref{pre:test}, this depends only on the function
field, and $u$ is extended additively to finite lists of fields. All
fourfolds in this section are the terminal Mori fibre spaces and
intermediate models in the ordinary Sarkisov paths supplied by
Theorem~\ref{bd:diagrams}, birational to the fixed rationally connected
fourfold. Every reference to an ordinary link below is to a link in
one of these chosen paths. In the eventual application that
fourfold is assumed rational. The common base of a link is denoted by
$Q$.
For a proper contraction $f:Y\to Z$ with the generic-fibre property in
Lemma~\ref{pre:mrc-fibration}, let $\operatorname{Div}_{Z}(Y)$ denote the
prime divisors on $Y$ whose images are proper subsets of $Z$.
We define
\begin{equation}
\label{sl:v-definition}
v(Y/Z)=
\sum_{E\in\operatorname{Div}_{Z}(Y)}u(E)
-\sum_{D\in\operatorname{Div}(Z)}u(D).
\end{equation}
This notation means a finite cancelled sum, as follows. There is a
dense open $U\subset Z$ such that, for every prime divisor $D$ of $Z$
meeting $U$, precisely one prime divisor $E_D$ of $Y$ dominates $D$,
and $E_D\dashrightarrow D$ has the generic-fibre property of
Lemma~\ref{pre:mrc-fibration}. To obtain $U$, spread a resolution of the
generic fibre, and shrink so that the resulting smooth proper family
has geometrically integral rationally connected fibres and the
birational map to the original family remains fibrewise birational
over dense opens. Generic flatness and openness of geometric
integrality give uniqueness of $E_D$ after a further shrinking.
Choose the same open so that $f$ is flat over $U$. By the dimension
formula, a prime divisor meeting $f^{-1}(U)$ and vertical over $U$
must dominate a divisor of $U$. It is therefore one of the divisors
$E_D$ just described.
Cancel $u(E_D)-u(D)=0$ for every such $D$.
Every remaining prime divisor of $Z$ is a component of $Z\setminus U$,
and every remaining prime divisor of $Y$ is a component of
$f^{-1}(Z\setminus U)$. Both are finite lists. Replacing $U$ by a
smaller good open only adds pairs of equal terms, so the resulting
integer is independent of $U$.
\begin{lemma}[Uniruled divisors in contraction fibres]
\label{sl:uniruled-vertical}
Let $f:Y\to Z$ be a projective contraction of normal varieties over an
algebraically closed field of characteristic zero. Suppose that $Y$
is of Fano type over $Z$. Let $E$ be a prime divisor which is a
component of $f^{-1}(f(E))$ and satisfies
$\dim E>\dim f(E)$. Then $E$ is uniruled.
Consequently:
\begin{enumerate}[label=(\roman*)]
\item if $\dim Y>\dim Z$, every divisor vertical over $Z$ is uniruled;
\item if $f$ is birational, every $f$-exceptional prime divisor is
uniruled.
\end{enumerate}
The same conclusions apply to geometric generic fibres of such a
contraction and to birational transforms of the divisors.
\end{lemma}
\begin{proof}
Choose a rational boundary $\Theta$ making $(Y,\Theta)$ klt and
$-(K_Y+\Theta)$ relatively ample. Then
$-K_Y=-(K_Y+\Theta)+\Theta$ is relatively big, and a sufficiently divisible
multiple of $-(K_Y+\Theta)$ is generated over $Z$.
Theorem~1.2 and Corollary~1.4 of \cite{HM07Vertical} give rational chain
connectedness of every fibre; the non-klt locus is empty.
The statement after passage to a characteristic-zero geometric generic
field follows by descent of the finite data and base extension.
Put $T=f(E)$ and choose a general point $e$ of $E$ outside the other
components of $f^{-1}(T)$. The fibre of $E\to T$ through $e$ has
positive dimension. A chain of rational curves in the fibre of $f$
joining $e$ to another point starts with a nonconstant rational curve
through $e$. That curve is contained in $E$: its image is in
$f^{-1}(T)$, and an irreducible curve through a point lying on only the
component $E$ must lie in that component. Thus rational curves cover
$E$, proving uniruledness. Equivalently, one can select a dominating
component of the parameter space of these rational curves. The
argument can be made after an uncountable algebraically closed
extension and descended, so it does not require an uncountability
assumption on the original field.
For (i), $f(E)$ is proper and $f^{-1}(f(E))$ is a proper closed subset
of the integral variety $Y$. Its component containing $E$ must equal
$E$, and
$\dim E-\dim f(E)\ge\dim Y-\dim Z>0$. The same component argument
works in (ii), where exceptionalness gives
$\dim E-\dim f(E)\ge1$. Finally uniruledness is birationally
invariant.
\end{proof}
The contractions $B\to Q$ which occur for the bases of link ends are
of Fano type over $Q$, by the base-contraction part of
Theorem~\ref{bd:diagrams}. Their geometric generic fibres are
geometrically integral and rationally connected; this also holds for
$V\to Q$ by composition. Thus all the following cancelled sums are
defined.
For a tower $V\to B\to Q$, let $h_Q(V/B)$ be the portion of
\eqref{sl:v-definition} consisting of divisors dominating $Q$:
\begin{equation}
\label{sl:h-definition}
h_Q(V/B)=
\sum_{\substack{E\in\operatorname{Div}_{B}(V)\\E\to Q\ {
dominant}}}u(E)
-\sum_{\substack{D\in\operatorname{Div}(B)\\D\to Q\ {
dominant}}}u(D).
\end{equation}
Use the same cancellation as for $v(V/B)$. A cancelled pair $E_D,D$
has the same dominance behaviour over $Q$, so this is well defined.
Partitioning both finite cancelled lists according to dominance over
$Q$ gives
\begin{equation}
\label{sl:tower}
v(V/Q)=v(V/B)-h_Q(V/B)+v(B/Q).
\end{equation}
Indeed a prime divisor of $V$ vertical over $Q$ is necessarily
vertical over $B$. In the right-hand side, the divisors of $B$
vertical over $Q$ cancel between the first two terms and $v(B/Q)$,
leaving exactly the defining sum for $v(V/Q)$. Good opens may be
shrunk simultaneously to perform these cancellations on finite lists.
\begin{lemma}
\label{sl:base-v-zero}
For every base contraction $B\to Q$ of an ordinary fourfold link,
$v(B/Q)=0$.
\end{lemma}
\begin{proof}
The assertion is immediate for an isomorphism. For a birational
contraction, nonexceptional prime divisors cancel with their
birational transforms on $Q$. The remaining divisors are uniruled by
Lemma~\ref{sl:uniruled-vertical}. Since $\dim B\le3$, these divisors
have dimension at most two, and their MRC quotients have dimension at
most one; they have test value zero.
If the contraction has positive relative dimension, then
$\dim Q\le2$. Every prime divisor of $Q$ has test value zero for
dimension reasons. Every divisor of $B$ vertical over $Q$ is
uniruled by Lemma~\ref{sl:uniruled-vertical}, and again has dimension
at most two, so its test value is zero. This includes $Q$ a point,
when there are no vertical divisors and no base divisors to sum.
\end{proof}
For a link $f:V/B\dashrightarrow V'/B'$ over $Q$, split its divisorial
cocycle into the contributions from valuations vertical and horizontal
over $Q$, as in \cite[Definition~2.3]{LinShinder26}:
\[
c_u(f)=c_{u,\mathrm{vert}/Q}(f)+c_{u,\mathrm{hor}/Q}(f).
\]
The signs are those of Lemma~\ref{pre:cocycle}: a divisor present
on the primed model and absent on the unprimed model contributes
positively. A common prime divisor has the same residue field and the
same dominance behaviour over $Q$ on both models.
\begin{proposition}[Vertical correction]
\label{sl:vertical}
For every ordinary fourfold link over $Q$ one has
\begin{align}
\label{sl:vertical-cocycle}
c_{u,\mathrm{vert}/Q}(f)&=v(V'/Q)-v(V/Q),\\
\label{sl:horizontal-residual}
c_u(f)-\bigl(v(V'/B')-v(V/B)\bigr)
&=c_{u,\mathrm{hor}/Q}(f)-h_Q(V'/B')+h_Q(V/B).
\end{align}
These identities hold for every test degree and require no
boundedness threshold.
\end{proposition}
\begin{proof}
In the difference $v(V'/Q)-v(V/Q)$ the base-divisor sums cancel.
Common vertical prime valuations cancel as well, leaving precisely
the signed exceptional divisors vertical over $Q$. This proves
\eqref{sl:vertical-cocycle}. Substitute \eqref{sl:tower} and use
Lemma~\ref{sl:base-v-zero} on both ends to obtain
\eqref{sl:horizontal-residual}.
\end{proof}
\subsection{The common-base dimensions other than two}
Denote the right-hand side of \eqref{sl:horizontal-residual} by
$H_Q(f)$. We next specify exactly where a degree threshold is needed.
\begin{proposition}
\label{sl:low-base}
There is an integer $d_{\mathrm{vert}}$, depending only on the bounded
marked families in Theorem~\ref{bd:diagrams} and the exclusions in
Theorem~\ref{bd:exclusions}, with the following property. For every
$d>d_{\mathrm{vert}}$, every allowed subcollection $\mathcal S$, and
every ordinary fourfold link $f$ whose common base satisfies
$\dim Q\le1$, one has
\[
c_{u,\mathrm{hor}/Q}(f)=h_Q(V/B)=h_Q(V'/B')=0.
\]
In particular $H_Q(f)=0$. If $\dim Q=3$, these equalities hold for
every $d$. The integer $d_{\mathrm{vert}}$ is independent of the link,
the birational map, the number of links in a path, and the choice of
$\mathcal S$.
\end{proposition}
\begin{proof}
Suppose first that $\dim Q=3$. Each Mori base has dimension at most
three and dominates $Q$, so its morphism to $Q$ is birational.
No divisor of either Mori base, and no divisor of $V$ vertical over
that base, dominates $Q$. Therefore both $h_Q$ terms vanish.
Normal proper generic curves over $\C(Q)$ are regular and hence
smooth in characteristic zero. A birational map between them is an
isomorphism. Their closed points are exactly the prime divisors of
the fourfolds dominating $Q$, so there are no horizontal divisorial
contributions either.
For $\dim Q=0$, the entire marked contraction diagram is bounded by
Theorem~\ref{bd:diagrams}. The exceptional prime divisors contributing
to $c_u(f)$ therefore have bounded birational models of dimension
three. Choose a marked good open for each contraction $V\to B$.
After cancellation, the divisors contributing to $h_Q(V/B)=v(V/B)$
are components of the marked bad locus on $B$ or its preimage on
$V$. These components have bounded birational models by the
\emph{marked-locus} assertion of Theorem~\ref{bd:diagrams}.
The bounded-variety exclusion in Theorem~\ref{bd:exclusions}\textup{(i)} gives a
single threshold beyond which every term has test value zero.
Now suppose $\dim Q=1$, and put $F=\C(Q)$. We use the bounded
diagrams after extension to $\overline F$. If $E$ is a horizontal
exceptional prime divisor of the link, the geometric generic fibre
components of $E\to Q$ are surfaces. On the side where the divisor
is extracted, each is exceptional for a divisorial contraction of
the generic threefold. They are uniruled by
Lemma~\ref{sl:uniruled-vertical}; any intervening small modifications
preserve their birational types. Their smooth birational models are
bounded by Theorem~\ref{bd:diagrams}. The threefold-over-a-curve
exclusion in Theorem~\ref{bd:exclusions}\textup{(iv)} therefore makes
$u(E)=0$ uniformly for large $d$. Here $u$ is evaluated on the original
complex threefold field $\C(E)$; only its geometric generic surface
components, rather than $E$ itself, are being placed in bounded families.
It remains to treat $h_Q(V/B)$, the other end being identical. Work
with a good open for the generic contraction
$V_F\to B_F$ and spread it over an open of $Q$. Shrinking $Q$ changes
no horizontal prime field. The cancellations defining $h_Q$ remove
all divisors meeting that good open. Consequently the geometric
generic components of the remaining prime divisors lie in the
bounded marked bad locus or its bounded preimage.
A remaining positive term is a divisor $P\subset V$ vertical over
$B$ and dominant over $Q$. Its geometric generic fibre components
are vertical divisors of $V_{\overline F}\to B_{\overline F}$.
They are uniruled surfaces by Lemma~\ref{sl:uniruled-vertical}, with
bounded birational models. Theorem~\ref{bd:exclusions}\textup{(iv)}
therefore gives $u(P)=0$ for large $d$. A remaining negative term is a divisor
$D\subset B$ dominant over $Q$. If $\dim D\le1$, its test value is
zero automatically. If $\dim D=2$, its geometric generic fibre
components over $Q$ are bounded curves. Were $u(D)=1$, the surface
$D$ would be birational to a member of $\mathcal S$; the induced
pencil, after Stein factorization, would have general fibre genus at
least $1+\sqrt d/2$. This contradicts the uniform genus bound for
those components when $d$ is large.
Only finitely many bounded families and types of marked components
have been used. Take the maximum of their exclusion thresholds.
All exclusions concern the full collection of degree-$d$
Picard-number-one K3 surfaces and therefore hold for every
subcollection $\mathcal S$. This proves the stated uniformity.
\end{proof}
\subsection{Reduction over a surface common base}
\begin{lemma}
\label{sl:generic-surface-reduction}
Suppose $\dim Q=2$. Then $k=\C(Q)$ is a rational function field in
two variables over $\C$. The generic fourfold-link models are smooth
projective geometrically integral surfaces over $k$, and all small
modifications in the link become isomorphisms. An end is either a
del Pezzo surface of Picard number one over $k$, or a conic bundle
over a smooth projective geometrically integral genus-zero curve
over $k$, of relative Picard number one. These statements do not
assert that the surface or the genus-zero curve is split over $k$.
In the conic case the total Picard number of the generic surface over
$k$ is two.
\end{lemma}
\begin{proof}
A smooth projective model of $Q$ is a dominant rational image of the
rationally connected fourfold, hence is rationally connected. A
smooth projective rationally connected complex surface is rational:
it is uniruled, its minimal model is ruled over a curve, and rational
connectedness forces that curve to have genus zero; see
\cite[Theorem~30]{AraujoKollar02} and \cite[Theorem~V.1]{Beauville78}. Consequently
$k\simeq\C(x,y)$.
A terminal fourfold is nonsingular in codimension two
\cite[Exercise~4.9.1.1]{KollarFlips057}, so its
singular locus has dimension at most one and cannot dominate $Q$.
The generic surface is regular, as a localization of its regular
total space near the generic fibre, and is smooth because $k$ is
perfect. The contractions have connected fibres and rationally
connected geometric generic fibres as above, giving geometric
integrality. Projectivity is inherited by base change. A small
birational map between smooth proper surfaces is an isomorphism:
the exceptional locus of a non-isomorphic birational morphism of
smooth surfaces contains a curve, and resolving a birational map
gives the same conclusion if it is non-isomorphic in either
direction. Such a curve would spread to a divisor exceptional for
the original small map.
If $\dim B=2$, the contraction $B\to Q$ is birational. The generic
end $V_k$ has ample anticanonical divisor and Picard number one over
$k$, hence is a del Pezzo surface. If $\dim B=3$, the generic base
$B_k$ is a normal proper curve with geometrically integral
rationally connected generic model, so it is a smooth genus-zero
curve. The generic morphism $V_k\to B_k$ has relatively ample
anticanonical divisor and relative Picard number one, hence is a
smooth-surface Mori conic bundle.
Here Picard numbers mean dimensions of numerical divisor spaces
\emph{over $k$}, not over its algebraic closure. To justify their
restriction from the fourfold, a divisor on $V_k$ closes up to a
Weil divisor on $V$, which is $\Q$-Cartier. This gives surjectivity
onto the generic numerical divisor space. A class numerically zero
over the indicated original base remains numerically zero on the
generic fibre: a generic curve and its intersection degrees spread
over an open of $Q$. Thus the relative rank is at most the original
rank one. The anticanonical class is nonzero, so it is exactly one.
The same argument bounds the numerical rank of the generic middle
surface by two. No passage to an algebraic closure is made in this
rank calculation.
In the conic case the base curve has numerical Picard number one,
its pullback to the surface is nonzero, and the relative numerical
divisor space has dimension one. Thus the total numerical divisor
space has dimension two.
\end{proof}
\begin{lemma}[The generic dichotomy]\label{sl:generic-dichotomy}
Suppose $\dim Q=2$ and put $k=\C(Q)$.
After identifying the middle generic surfaces by the restricted small
maps, let $W$ be their common smooth model. Exactly one of the following
reductions applies:
\begin{enumerate}[label=\textup{(\roman*)}]
\item The restricted link induces an isomorphism of the structured
generic ends. In the conic case their curve fields are the same
embedded subfield of $k(W)$. Its horizontal cocycle and $H_Q$ vanish.
\item The two ends are obtained from $W$ by distinct elementary
$K_W$-negative contractions, each either divisorial to a rank-one del
Pezzo surface or a conic contraction. Here $\rho(W/k)=2$, the two
contractions give the two extremal rays, and $-K_W$ is ample.
\end{enumerate}
In particular two distinct conic structures on the same surface belong
to \textup{(ii)}, not \textup{(i)}.
\end{lemma}
\begin{proof}
By Lemma~\ref{sl:generic-surface-reduction}, $\rho(W/k)\leq2$.
If a divisorial contraction $W\to S$ survives on the generic fibre,
then $\rho(W/k)=\rho(S/k)+1$. The end $S$ therefore has rank one and
is a del Pezzo surface. Extraction from a conic end, which has rank
two, would force $\rho(W/k)\geq3$. Thus any surviving extraction forces
$\rho(W/k)=2$. If the opposite end has no surviving extraction, its
surface is $W$ itself and cannot be a rank-one del Pezzo. It must
instead carry its conic contraction. If both ends have surviving
extractions, both are divisorial contractions from $W$ to rank-one
del Pezzo surfaces.
If neither side has a surviving extraction, both generic end surfaces
are $W$. Rank one then gives two structures over a point, and the
restricted map is a structured isomorphism. Rank two gives two conic
contractions. These may agree or may be distinct; the latter case is
retained as a two-conic move.
In all the rank-two cases, compare the two end contractions on $W$.
If they contract the same extremal ray, uniqueness of the contraction
identifies their normal targets. Indeed, each is constant on the
connected fibres of the other, hence factors through it, and the two
factorizations are inverse. For conic contractions this identifies
their pullback curve fields inside $k(W)$, not merely the abstract
isomorphism classes of the curves. For identical birational
contractions the common exceptional orbit cancels from the horizontal
cocycle. Thus the restricted birational map is a structured
isomorphism in either case, with equal vertical divisor data and
$H_Q=0$.
Otherwise the two contractions give distinct $K_W$-negative extremal
rays. These are the two boundary rays of the two-dimensional cone of
curves of the projective surface $W$. The anticanonical divisor is
positive on both and is therefore ample. This proves \textup{(ii)}.
The argument uses numerical classes and contractions over $k$ and
does not require the generic surfaces or conic bases to split.
\end{proof}
\subsection{Finite sets attached to the generic surfaces}
\label{sl:generic-sets}
We now work over the rational surface field $k=\C(Q)$ obtained when
$\dim Q=2$. Write $\overline k$ for an algebraic closure and
$G_k=\operatorname{Gal}(\overline k/k)$. All finite $G_k$-sets below carry
continuous actions. Extend the test to these sets by
\begin{equation}\label{sl:orbit-test}
u(A)=\sum_{O\in A/G_k}u(k(O)),
\end{equation}
where $k(O)$ is the finite extension corresponding to the transitive orbit
$O$. These are surface function fields over $\C$. Blowing up a closed point
with residue field $K$ adds a divisor with field $K(t)$, so its horizontal
contribution is exactly $u(K)$ by Lemma~\ref{pre:mrc-fibration}.
We will repeatedly use
\begin{equation}\label{sl:small-orbits}
|O|\leq2\quad\Longrightarrow\quad u(k(O))=0.
\end{equation}
This is the small-orbit observation of Lemma~\ref{pre:small-orbits}:
$k$ is rational, and a quadratic extension has a nonidentity
$k$-automorphism, excluded for a test K3 field.
The generic Mori surfaces are rank-one del Pezzo surfaces over $k$, or
relatively rank-one conic bundles $S\to C$ over a geometrically integral
genus-zero curve. We do not assume that either the surfaces or the curve
$C$ are split over $k$.
\begin{lemma}[Conic fibres and the horizontal vertical correction]
\label{sl:conic-h}
For a conic-bundle generic surface $S\to C$, let $\Delta$ be the finite
$G_k$-set of singular geometric fibres and let $\widetilde\Delta$ be the
double set of their components. Each component cover over a discriminant
orbit is nonsplit. The quaternion class of the generic conic over $k(C)$
ramifies exactly at these points, and its residue extensions are exactly
these component covers. Moreover,
\begin{equation}\label{sl:h-discriminant}
h_Q(V/B)=u(\widetilde\Delta)-u(\Delta)=-u(\Delta).
\end{equation}
For a rank-one del Pezzo generic surface, $h_Q(V/B)=0$.
\end{lemma}
\begin{proof}
The morphism from the smooth surface to the smooth curve is flat,
so its fibres have the arithmetic genus of the generic conic.
Over $\overline k$, every fibre is connected, has arithmetic genus zero,
and has anticanonical degree two. The anticanonical degree of each
irreducible component is a positive integer. An irreducible reduced
fibre therefore has arithmetic genus zero and is a smooth rational curve.
A double fibre $2R$ would have $R^2=0$ and $K_S\cdot R=-1$, contradicting
adjunction parity. The only remaining possibility is a reduced pair
$R_1+R_2$ with $-K_S\cdot R_i=1$. Put $m=R_1\cdot R_2$. Since
$(R_1+R_2)\cdot R_i=0$ and the fibre is connected, adjunction gives
\[
2p_a(R_i)-2=-m-1,\qquad m\geq1.
\]
Thus $m=1$, both curves are smooth rational curves of self-intersection
$-1$, and they meet transversely.
If the component cover over an orbit were split, the sum of one chosen
component over each point of that orbit would descend to a divisor on
$S$. It has zero intersection with a general fibre and negative
intersection with every component in that chosen orbit. This is
impossible when the relative numerical divisor space is one-dimensional:
the general fibre is a nonzero relative curve class and spans its dual.
Consequently every component orbit has field $L$ quadratic over the
corresponding discriminant field $K$.
The relative anticanonical system embeds the fibres as plane conics:
on a smooth fibre it has degree two, and on a reducible fibre it has
degree one on each component, with vanishing first cohomology. Cohomology
and base change give the local rank-three system. At a singular fibre
the total surface is regular, so after diagonalizing the quadratic form
over the local discrete valuation ring, its nonsingular binary part has
unit determinant and the remaining coefficient has valuation one.
More explicitly, with uniformizer $t$ the equation is
\[
ax^2+by^2+tc z^2=0,\qquad a,b,c\text{ units}.
\]
An exponent of $t$ greater than one would make the total space singular
at the geometric node. We use the Kummer identification and quaternion
residue formula of \cite[\S2, equations~(1)--(4)]{AuelBohningBothmerPirutka20}.
The associated quaternion can be written
$(-ab,-act)$; its tame residue is the square class of $-ab$ in the
residue field. This is exactly the extension separating the two lines
$ax^2+by^2=0$. At a smooth fibre all three diagonal coefficients are
units, so the residue vanishes.
The corresponding vertical divisor has function field $L(t)$, since a
line is split over its component field. The involution of $L/K$ excludes
$u(L)=1$ by the automorphism clause of Lemma~\ref{pre:small-orbits};
the degree of $L$ over $k$ need not be two. All smooth
fibres cancel against their base points by the MRC rule. This proves
\eqref{sl:h-discriminant}. On a del Pezzo side the generic base is a
point, so there are no vertical primes dominating $Q$.
\end{proof}
We have computed $h_Q$ for both kinds of generic end. It remains to
express the horizontal cocycle as an endpoint difference. The following
finite $G_k$-sets will define a candidate function $w$. Their orbit fields
matter: equality of rational permutation representations alone need not
identify the fields tested by $u$.
We use the geometric classification, intersection lattices and
anticanonical models of smooth del Pezzo surfaces
\cite[Sections~3.4 and~6.1]{DolgachevIskovskikh2009}.
All plane blowup bases below are chosen over $\overline k$.
For a del Pezzo surface $S$, let
\[
A(S)=\{F\in\operatorname{Pic}(S_{\overline k}):
F^2=0,\ -K_S\cdot F=2\}.
\]
For a conic bundle, let $P_\Delta$ be the set of all choices of one
component in each singular geometric fibre. It has $2^{|\Delta|}$
elements, with its natural $G_k$-action.
The del Pezzo pencil sets and the endpoint-difference calculation below
adapt the virtual N\'eron--Severi viewpoint of Lin, Shinder and Zimmermann
\cite[Definition~5.2 and Proposition~5.5]{LinShinderZimmermann23}.
The conic correction also records the component choices needed for the
generic Mori surfaces here.
The two-ray calculation below will show why only the
degree-five and degree-six pencil sets, and the component choices
over three singular conic fibres, enter this function. Once that
identity is proved, \eqref{sl:horizontal-residual} will express the
remaining link contribution as the change of $w-h_Q$.
\begin{definition}\label{sl:w-def}
For a generic Mori surface with its specified structure, set
\[
w(S)=
\begin{cases}
u(A(S)),&S\text{ is a rank-one del Pezzo of degree }5\text{ or }6,\\
\tfrac12u(P_\Delta),&S\to C\text{ is a conic bundle and }|\Delta|=3,\\
0,&\text{otherwise}.
\end{cases}
\]
The del Pezzo value depends on the surface; the conic value includes its
chosen genus-zero curve structure.
\end{definition}
For a fourfold node $V\to B$ with $\dim B=2$, write
$w(V/B):=w(V_\eta)$, where $\eta=\Spec\C(B)$.
The factor $1/2$ is harmless even integrally. Complementing all choices
is a fixed-point-free, equivariant involution of $P_\Delta$. An orbit
preserved by this involution has a nonidentity field automorphism and
contributes zero. The remaining orbits occur in isomorphic pairs.
Thus $u(P_\Delta)$ is even.
\begin{lemma}[The pencil sets]\label{sl:pencil-sets}
The sets $A(S)$ have respectively five and three elements in degrees
five and six. In degree six there is also a two-element set
\[
A_2(S)=\{C\in\operatorname{Pic}(S_{\overline k}):
C^2=1,\ -K_S\cdot C=3\},
\]
and an equality of rational $G_k$-representations
\begin{equation}\label{sl:degree-six-representation}
\operatorname{Pic}(S_{\overline k})_\Q\oplus\Q
\simeq\Q[A(S)]\oplus\Q[A_2(S)].
\end{equation}
If $S$ has Picard rank one over $k$, both finite sets in
\eqref{sl:degree-six-representation} are transitive.
\end{lemma}
\begin{proof}
Use a plane blowup basis $L,E_1,\ldots,E_n$ with $n=9-K_S^2$.
A class $xL-\sum y_iE_i$ belongs to $A(S)$ exactly when
\[
\sum y_i=3x-2,\qquad \sum y_i^2=x^2.
\]
Cauchy's inequality gives $(3x-2)^2\leq nx^2$. For $n=4$ this restricts
the integer $x$ to $1,2$. At $x=1$ there is one coefficient $y_i=1$
and all others vanish; at $x=2$ all four coefficients are one. These
are the four line pencils and the conic pencil. For $n=3$ the inequality
forces $x=1$, giving the three line pencils. The same calculation for
$A_2(S)$ gives $x=1,2$ and the two classes
\[
L,\qquad 2L-E_1-E_2-E_3.
\]
The three pencil classes $L-E_i$ and these two plane classes span the
degree-six Picard space. Their only linear relation is the equality
of the two class sums, each equal to $-K_S$. The relation is invariant
under $G_k$; semisimplicity yields \eqref{sl:degree-six-representation}.
Taking invariants gives
\[
2=\#(A(S)/G_k)+\#(A_2(S)/G_k).
\]
The Brauer obstruction to descending an invariant line bundle is torsion
\cite[Remark~2.7]{AuelBernardara2018}. Thus invariant rational Picard
classes descend, and the invariant Picard space has dimension one. Both finite
sets are nonempty, proving transitivity.
\end{proof}
\subsection{The two-ray surface calculation}
\label{sl:two-ray-calculation}
\begin{proposition}[Surface-link identity]\label{sl:surface-identity}
Let $\varphi:V/B\dashrightarrow V'/B'$ be an ordinary fourfold link over a
two-dimensional common base $Q$. Put $k=\C(Q)$, and denote its
generic end surfaces, with their induced Mori structures, by $S$ and
$S'$. If its restriction is the nontrivial two-ray move in
Lemma~\ref{sl:generic-dichotomy}\textup{(ii)}, then
\begin{equation}\label{sl:surface-equation}
c_{u,\mathrm{hor}/Q}(\varphi)=w(S')-w(S).
\end{equation}
Every conic surface participating in such a move has at most seven
singular geometric fibres. In the other case of
Lemma~\ref{sl:generic-dichotomy}, $H_Q=0$ without any bound on the
discriminant size.
\end{proposition}
\begin{proof}
By Lemma~\ref{sl:generic-dichotomy}, the common smooth surface $W$ is
del Pezzo with a two-dimensional invariant numerical plane and the
two end contractions generate its two rays. Each is either divisorial
to a smooth surface or a conic contraction. For a divisorial
contraction, pass first to $\overline k$. Its exceptional intersection
matrix is negative definite. An exceptional component $R$ has
$R^2<0$ and $K_W\cdot R<0$; adjunction
$2p_a(R)-2=R^2+K_W\cdot R$ forces both intersections to equal $-1$
and $R$ to be a smooth rational curve. Two such curves must be
disjoint, since otherwise their two-by-two intersection matrix would
not be negative definite. Thus the contraction is the blowdown of
disjoint $(-1)$-curves to distinct smooth points. Each Galois orbit
of these curves contributes an independent invariant relative divisor
class. Relative Picard number one therefore gives a single orbit.
The contraction is consequently the blowup of one closed point over
$k$, and the equivariant correspondence between the geometric points
and their exceptional curves identifies their orbit fields.
Put $D=K_W^2>0$.
Every fibre class used below is the integral class of a geometric
fibre in $\operatorname{Pic}(W_{\overline k})$. It is $G_k$-invariant,
has square zero and anticanonical degree two, and lies in the rational
numerical plane over $k$ by the torsion descent argument used in
Lemma~\ref{sl:pencil-sets}. A closed fibre over a point of degree $r$
on the original genus-zero curve has class $r$ times this geometric
class. Thus none of the following calculations requires a $k$-point
on the conic base.
For a divisorial end let $E$ denote its orbit of exceptional curves,
write $e=|E|$, and put $E_\Sigma=\sum_{R\in E}R$ on
$W_{\overline k}$. Thus $u(E)=u(k(E))$ counts one orbit field,
whereas $E_\Sigma^2=-e$ counts geometric curves in an intersection
calculation. When both ends are divisorial, use $F$, $f=|F|$, and
$F_\Sigma$ for the corresponding data at the primed end. In the diagram
\[
\begin{tikzcd}[column sep=large]
&W\arrow[dl,"\pi"']\arrow[dr,"\pi'"]&\\
S\arrow[rr,dashed,"\varphi_k"']&&S'
\end{tikzcd}
\]
$E$ is the exceptional set of $\pi$ and $F$ that of $\pi'$.
The factorization $\varphi_k=\pi'\circ\pi^{-1}$ therefore contributes
$u(E)-u(F)$ by Lemma~\ref{pre:cocycle}: the extraction has positive
sign and the contraction negative sign. Neither term is multiplied
by its orbit size.
\paragraph{One divisorial end.}
Let $\pi:W\to S$ be the contraction to the rank-one del Pezzo end,
of degree $a$, and let $W\to C$ be the other, conic contraction.
Then $D=a-e>0$. Write
$H=\pi^*(-K_S)$. The plane spanned by $H$ and $E_\Sigma$ has
intersection form $H^2=a$, $H\cdot E_\Sigma=0$, and
$E_\Sigma^2=-e$. A nonzero isotropic fibre class in this rational
plane therefore gives $ax^2=ey^2$ with nonzero rational $x,y$, so
$ae$ is a square. Since $1\leq e<a\leq9$, the
complete list is
\begin{equation}\label{sl:div-conic-table}
(a,e)=(4,1),\ (8,2),\ (9,1),\ (9,4).
\end{equation}
Indeed, two positive integers with square product have the same
squarefree part. Below ten the only unequal such pairs with smaller
entry less than the larger are $1,4$, $2,8$, $1,9$, and $4,9$.
Over $\overline k$, contracting one line in every singular conic fibre
gives a ruled surface over $\mathbf P^1$. Its canonical square is eight,
so
\begin{equation}\label{sl:conic-size}
|\Delta|=8-D.
\end{equation}
The first three cases of \eqref{sl:div-conic-table} have extraction
degree at most two and discriminant size different from three. Their
two $w$ values and their cocycle contributions vanish.
In the remaining case $(a,e)=(9,4)$, work geometrically on the plane
blown up at four points. Galois fixes $L$ and permutes the four
exceptional classes transitively. Thus the invariant fibre class
has the form $F=xL-y\sum_{i=1}^4E_i$ with $x,y\in\Z$. The equations
$F^2=0$ and $-K_W\cdot F=2$ give $x=2$, $y=1$.
The conic pencil is therefore the pencil through the four points.
Its three reducible members correspond to the partitions of these
points into two pairs. For each point $p$, select in every partition
the pair containing $p$. This gives a star of three lines; selecting
the other pair gives its complementary triangle. The four stars
and four triangles exhaust the eight possible selections. The
construction commutes with every permutation of the four points,
so there is an isomorphism of $G_k$-sets
\begin{equation}\label{sl:eight-orientations}
P_\Delta\simeq E\sqcup E.
\end{equation}
Consequently $w(W\to C)=u(E)$, which is the extraction contribution.
Reversing the link reverses both signs.
\paragraph{Two conic ends.}
The same surface has two fibre classes $F_1,F_2$. Put $m=F_1\cdot F_2$.
They are distinct isotropic rays, so $m>0$, and their anticanonical
degrees are two. In their numerical plane,
\[
-K_W=\frac2m(F_1+F_2),\qquad
D=\frac8m,\qquad F_1+F_2=\frac4D(-K_W).
\]
Since $m$ is a positive integer, the equality $D=8/m$ excludes degree five.
Thus neither conic bundle has discriminant size three, by
\eqref{sl:conic-size}. Their $w$ values vanish and there is no
divisorial contribution.
\paragraph{Two divisorial ends: the numerical list.}
Now let $e,f$ be the orbit sizes, so that the end degrees are $D+e$
and $D+f$, each at most nine. If $D=1$ or $2$, the Bertini or Geiser involution $\iota$ fixes $K_W$
and acts as $-1$ on $K_W^\perp$
\cite[Sections~6.6 and~6.7]{DolgachevIskovskikh2009}.
It is defined over $k$: the corresponding complete anticanonical or
bi-anticanonical system is defined over $k$, and the nontrivial
involution of its quadratic function-field extension extends to $W$.
If $\iota$ preserved an exceptional ray with orbit sum $E_\Sigma$, then
$\iota_*E_\Sigma=\lambda E_\Sigma$ for some $\lambda>0$; intersection with $-K_W$
forces $\lambda=1$. The fixed-space formula would then give
$E_\Sigma\in\Q K_W$, contrary to $E_\Sigma^2<0<K_W^2$.
Thus $\iota$ exchanges the two contractions and identifies their entire
exceptional $G_k$-sets and output surfaces over $k$. Both sides of
\eqref{sl:surface-equation} vanish.
For the remaining degrees we first restrict the possible orbit sizes by
intersection theory, then identify the finite $G_k$-sets. The target is
\[
u(E)-u(F)=w(S')-w(S).
\]
Small orbits contribute zero; the other terms will be matched through
equivariant bijections of exceptional or pencil sets.
Suppose $D\geq3$ and put $m=E_\Sigma\cdot F_\Sigma$. The Gram determinant of
$-K_W,E_\Sigma,F_\Sigma$, which lie in a plane, gives
\[
Dm^2-2efm-ef(D+e+f)=0.
\]
Since the two effective exceptional sums meet nonnegatively, the
solution is
\begin{equation}\label{sl:intersection-formula}
m=\frac{ef+\sqrt{ef(D+e)(D+f)}}D.
\end{equation}
Each curve in an orbit has the same total incidence with the opposite
orbit. Thus $m/e$ and $m/f$ are integers. The anticanonical embedding
realizes $(-1)$ curves as lines; two different such curves meet at most
once. In particular $m\leq ef$.
These conditions give precisely the following necessary possibilities,
with $e\leq f$:
\begin{center}
\begin{tabular}{@{}cl@{}}
\toprule
$D$&$(e,f,m)$\\
\midrule
3&$(2,5,10),\ (3,3,9),\ (6,6,30)$\\
4&$(1,5,5),\ (2,2,4),\ (4,4,12)$\\
5&$(1,3,3)$\\
6&$(3,3,6)$\\
7&$(1,2,2)$\\
\bottomrule
\end{tabular}
\end{center}
Here is a direct exhaustion. For $e=f$, formula
\eqref{sl:intersection-formula} gives $m/e=1+2e/D$, so $D\mid2e$;
impose also $1\leq e\leq9-D$ and $m\leq e^2$. This gives exactly
the diagonal entries displayed. For $e<f$, the values $i(i+D)$ for
$1\leq i\leq9-D$ must have matching squarefree parts. The lists for
$D=3,4,5,6,7$ are respectively
\[
\begin{array}{c|l}
3&4,10,18,28,40,54\\
4&5,12,21,32,45\\
5&6,14,24,36\\
6&7,16,27\\
7&8,18.
\end{array}
\]
They give only $(2,5),(1,5),(1,3),(1,2)$ in their respective rows.
For $D\geq8$ the permitted range has at most one entry and supplies
neither a diagonal nor an off-diagonal case.
\paragraph{Identifying the finite sets.}
In the cases $(D,e,f)=(3,2,5)$ and $(3,3,3)$, every curve in $F$ meets
every curve in $E$ once. After contracting $E$, its image class $C$
on $S$ has
\[
(K_S^2,C^2,-K_S\cdot C)=(5,1,3)
\quad\text{or}\quad(6,2,4),
\]
respectively. Therefore $-K_S-C$ is in $A(S)$. This construction is
injective: equal image classes have equal pullbacks, and all the
multiplicities at the extracted points equal one, so their strict
transforms have the same $(-1)$ class. A $(-1)$ class has a unique
effective representative. The cardinalities in Lemma~\ref{sl:pencil-sets}
then show that
\[
F\simeq A(S)
\]
as actual $G_k$-sets. For $(3,3,3)$ the construction on the other
side also gives $E\simeq A(S')$. For $(3,2,5)$ the other side has
degree eight and $u(E)=0$. Thus the desired difference is $u(E)-u(F)$
in both cases.
For $(D,e,f)=(4,1,5)$ and $(5,1,3)$ every opposing curve meets the
single extracted curve once. Its image already has square zero and
anticanonical degree two, so the image itself identifies $F$ with
$A(S)$. The injectivity argument is unchanged. The opposite end has
degree nine or eight, respectively, and the one-point orbit contributes
zero. These cases again give \eqref{sl:surface-equation}.
In the diagonal cases $(D,e)=(3,6),(4,4),(6,3)$, the value of $m/e$
is $e-1$. Each curve therefore misses exactly one curve in the opposite
orbit. This rule gives an equivariant bijection $E\simeq F$, since
the same statement holds in both directions. The output degrees are
nine or eight, so no pencil terms occur. In degree seven both orbit
sizes are at most two and all terms vanish by \eqref{sl:small-orbits}.
It remains to treat $(D,e,f)=(4,2,2)$, whose two outputs have degree
six. Here we must identify the actual pencil $G_k$-sets, since an
equality of rational permutation representations alone need not
identify their orbit fields. The Picard blowup decompositions imply
\[
\operatorname{Pic}(S_{\overline k})_\Q\oplus\Q[E]
\simeq
\operatorname{Pic}(S'_{\overline k})_\Q\oplus\Q[F].
\]
Insert \eqref{sl:degree-six-representation} on both sides. The
permutation representations of $E,F,A_2(S),A_2(S')$ have only
one-dimensional rational constituents, because all four sets have
two elements. By transitivity, the reduced representation of each
triple $A(S)$ or $A(S')$ is irreducible over $\Q$: its image is either
$C_3$, acting through $\Q(\zeta_3)$, or $S_3$, acting through its
standard representation. Semisimplicity consequently identifies
these two reduced representations. Both are faithful on their
image, so their kernels in $G_k$ agree. The quotient is $C_3$ or
$S_3$; its transitive action on three points is unique up to
isomorphism (the stabilizer is trivial or an order-two subgroup,
and all such subgroups of $S_3$ are conjugate). Hence
\[
A(S)\simeq A(S')
\]
as $G_k$-sets. The two extraction orbits contribute zero by
\eqref{sl:small-orbits}, completing the final case.
Finally, positivity of $D$ in every nontrivial conic case and
\eqref{sl:conic-size} give $|\Delta|\leq7$. In the structured-isomorphism
case of Lemma~\ref{sl:generic-dichotomy}, the horizontal cocycle is
zero and the $h_Q$ values agree; this case includes cancellation of
identical generic end contractions. Thus $H_Q=0$ even when the conic
discriminant is arbitrarily large.
\end{proof}
\begin{remark}\label{sl:local-correction}
Combining Proposition~\ref{sl:surface-identity} with
Proposition~\ref{sl:vertical}, the residual on a nontrivial generic
surface link is the change of $w-h_Q$. On a del Pezzo side this is
$w$; on a conic side it is
\[
u(\Delta)+\mathbf1_{|\Delta|=3}\,\tfrac12u(P_\Delta).
\]
The surface field $k$ is part of this formula. Its compatibility
across different two-dimensional common bases is proved in the next
section. The local calculation alone does not assert that this is
already an invariant of the entire conic Mori fibre space.
\end{remark}
This interpretation agrees with the virtual N\'eron--Severi sets
of Lin--Shinder--Zimmermann
\cite[Proposition~4.8, Definition~5.2 and Proposition~5.5]{LinShinderZimmermann23}:
their point and two-element terms vanish under our test. The explicit
proof was included to retain the actual orbit fields at every step.
==== END Irrational-cubic-fourfolds-with-geometric-K3-categories//surface-links.tex ====
==== BEGIN Irrational-cubic-fourfolds-with-geometric-K3-categories//references.bib ====
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==== END Irrational-cubic-fourfolds-with-geometric-K3-categories//references.bib ====