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The concrete question concerns a very common kind of map in
algebraic geometry, and it answers it with one concrete computation. The map
takes a space X down to a smaller space Z; the fibres have trivial
canonical class in a suitable sense; and the question is how badly the base
piece of the canonical bundle formula can be written when you are allowed to
fiddle with the representative. The answer says: not badly at all, as long as
the total dimension is at most four and the coefficients at the boundary are
drawn from a fixed finite set. Every quantity that the formula produces then
has a denominator bounded by a constant that knows only the dimension and the
coefficient set, never the individual map.

There are three intended readers. The first is a graduate student in
algebraic geometry who has met the minimal model program and the canonical
bundle formula but who has not yet seen this uniformity argument. The second
is a strong undergraduate who wants the shape of the argument and is willing
to take some of the deep input theorems on faith. The third is a curious
high-school reader who wants to understand what is being claimed and why it is
hard. Sections~sec:question and~sec:vocab, and the opening words of
every later section, are written for the third audience: they say in plain
language what is about to be proved before any formula appears. Everything
after those opening words is written at the first audience's level.

Two registers run throughout. There is a plain form, one to
five sentences, that a reader can take in without stopping. There is a
technical form, which is the actual statement or proof. The two
describe the same object. If the technical form is out of reach, the plain
form still tells you what the object is and what it is for.

Two features shape the result: it is a uniformity (not existence) theorem, and
it uses that the total dimension is at most four. Sections~sec:question
through~sec:residue prove the main theorem; Sections~sec:effective
and~sec:torsion apply it.



medskipnoindentIn words. Take a compact space X that maps onto a smaller space
Z, and suppose the fibres have ``trivial canonical class''. Then a classical
theorem says that the canonical class of X is the pullback of a class on
Z plus a correction that measures how singular the fibres are. That
correction has two pieces: a boundary divisor that records which fibres are
badly singular, and a ``moduli'' part that records something finer, the
one-parameter family of fibres through each point of Z. The moduli part is
only defined up to adding a principal divisor from the base, and adding a
principal divisor can change its coefficients by nasty fractions. The question
of the argument is whether, for a fixed dimension bound and a fixed set of
allowed boundary coefficients, one can always choose the representative so
that those fractions have a common, bounded denominator. The answer is yes,
and the bound knows nothing about the map.

medskipnoindentTechnical setting. Let (X,B) be a projective log canonical pair
over C of dimension d, with B an effective boundary whose coefficients
lie in a fixed finite set Phisubset[0,1]capQ. Let
[
 f: X→ Z
]
be a contraction: a surjective projective morphism of normal projective
varieties with f_*OO_X=OO_Z. Assume dim Z>0 and that
[
 K_X+Bsim_{Q} f^*D
]
for a rational Cartier divisor D on Z. Then the adjoint of the generic
fibre is rationally linearly trivial, and (X,B) is a log Calabi--Yau
fibration over Z.

medskipnoindentThe canonical bundle formula. On a suitable smooth base model
q: Wto Z one writes
[
 K_X+Bsim_{Q} f^*(K_Z+B_Z+M_Z),
]
where B_Z is the discriminant: at a prime Psubset Z its coefficient
is 1-t_P, with t_P the log canonical threshold of f^*P over the generic
point of P. The remaining piece M is the moduli b-divisor.
It is a b-divisor rather than a divisor because it only descends to a divisor
M_W on some high birational model Wto Z, and its pullbacks to higher
models are declared. When M_W is nef one says M is b-nef;
when pM_W is Cartier one says pbfM is b-Cartier.

The defect is the phrase ``on a suitable model''. Change D to a different
divisor D_Z in the same rational linear class, and M_W changes by a
principal divisor that comes from the base. That can change denominators
arbitrarily. The point of an exact presentation is to freeze the choice
of D_Z so that the moduli part has a denominator controlled from outside.

medskipnoindentPlain form of the main theorem. Fix the dimension dle 4 and the
coefficient set Phi. There is a single integer p, and a DCC set
B of allowed discriminant coefficients, both depending only on d
and Phi, with the following property. For every log Calabi--Yau fibration
of this dimension and coefficient set there is a corrected principal divisor
(1/p_0)Div(psi) and a representative D_Z of the base class such that
[
 K_X+B+frac1{p_0}Div(psi)=f^*D_Z
]
holds as an equality of actual rational divisors, with
D_Z=K_Z+B_Z+M_Z, with B_Z effective and coefficients in B, with
(Z,B_Z+M_Z) generalized log canonical, and with pbfM b-Cartier. The
integer p_0 divides p and clears the denominators of Phi. The bound p
does not depend on the fibration, only on d and Phi.

medskipnoindentTechnical form.

[Uniform relative denominators]
Fix a finite set Phisubset[0,1]capQ and an integer 1le dle 4.
There are positive integers p_0mid p, with p_0 clearing Phi, and a
rational DCC set B[0,1], both depending only on d and
Phi, with the following property. Suppose (X,B) is a projective log
canonical pair of dimension d with coefficients in Phi, and
[
 f: X→ Z, dim Z>0, K_X+Bsim_{Q} f^*D,
]
where f is a contraction and D is a rational Cartier divisor. Then there
exist psiinC(X)^* and D_Zsim_{Q}D such that

 K_X+B+frac1{p_0}Div(psi)=f^*D_Z, D_Z=K_Z+B_Z+M_Z,

where B_Z is effective with coefficients in B, the pair
(Z,B_Z+M_Z) is generalized log canonical, and M is b-nef with
pbfM b-Cartier. If (X,B) is klt then the generalized pair is
generalized klt.


medskipnoindentWhy the two integers are separate. The integer p_0 does two
jobs: it lets the principal correction psi produce an exact pullback, and on
the generic fibre it is a trivializing degree for the log Calabi--Yau pair
there. The integer p is larger: it clears the moduli trace on a model that
determines M.M. The discriminant is controlled separately, by the DCC
set B.

medskipnoindentWhat is genuinely new. Qualitative versions of the formula put the
moduli part on a model and make it nef, but the index may depend on the
fibration (Ambro, Fujino--Gongyo). The curve case of Floris even shows that,
when the generic fibre is a rational curve and the fibre index is allowed to
vary, no polynomial bound on the clearing integer can exist. The theorem says
that when the total dimension is at most four and the boundary
coefficients are from a fixed finite set, the index can be chosen uniformly.
The source of the uniformity is the index theorem for the reduced special
fibre, whose dimension is one less than the total.

{small./pdf/unknown-nd-math0301180-small.pdf}



A short glossary follows. Each item is one or two sentences; a reader who
already knows the term can skip it. Everything later refers back by item
number. Where a term has a long standard definition only the shape is recorded,
not the full definition, and point the reader to Koll'ar--Mori or the Stacks
Project.
[(V1)] A variety is an integral separated scheme of finite type over
C. Unless stated otherwise all varieties here are normal and, in the global
statements, projective. Normal means the local rings are normal domains;
a normal variety is smooth outside a closed set of codimension at least two.
A resolution is a proper birational morphism from a smooth variety.
[(V2)] A contraction f: Xto Z is a surjective projective
morphism of normal projective varieties with f_*OO_X=OO_Z. Equivalently, the
function field C(Z) is algebraically closed in C(X), so the generic fibre
is geometrically integral and the fibres are connected.
[(V3)] A divisor on a normal variety is a formal finite Z-linear
combination of prime divisors. A Q-divisor allows rational
coefficients. A divisor is effective when all coefficients are
nonnegative. A boundary is an effective Q-divisor; the boundary
coefficients are its coefficients.
[(V4)] Two Q-divisors are linearly equivalent, written sim,
when their difference is the divisor of a rational function. They are
rationally linearly equivalent, written sim_{Q}, when a positive
integer multiple of the difference is. They are numerically equivalent,
written equiv, when they meet every complete curve the same way. A divisor
is big, nef, or pseudoeffective when its class has the
corresponding positivity: big means it grows like an ample class on high
multiples, nef means nonnegative degree on every curve, pseudoeffective means it
lies in the closure of the cone of effective classes.
[(V5)] The canonical divisor K_X of a normal variety is the
divisor of a rational top form; on a smooth variety it is the determinant of
the cotangent bundle. It is defined up to linear equivalence. A log pair
(X,B) is a normal variety together with a boundary.
[(V6)] For a prime divisor E over X (that is, on some resolution
g: Yto X with E exceptional or strict transform), the
discrepancy of E with respect to (X,B) is the coefficient of E in
K_Y-g^*(K_X+B) when g is chosen so that this pullback makes sense; it is the
discrepancy when K_Y and K_X are fixed compatibly. The pair is
log canonical (lc) when every discrepancy is at least -1, and
klt when every discrepancy is strictly bigger than -1. The pair is
dlt when it is lc and the coefficients are at most one and the crossings
are simple normal; dlt is a convenient regular representative of an lc pair.
[(V7)] The pair is slc, semi-log-canonical, when it is defined on a
possibly nonnormal variety with pure-dimension components, is S_2, is nodal in
codimension one, and its normalization with the conductor included is lc. This
is the right singular class on a reduced fibre that may be reducible and
nonnormal.
[(V8)] The log canonical threshold lct(X,B;P) of a prime P
over a point of Z is the largest tge0 such that the pair
(X,B+t,f^*P) is lc over that point. In the canonical bundle formula the
discriminant coefficient at P is 1-t_P. A threshold is a single rational
number, so it is the natural quantity to try to control uniformly.
[(V9)] A b-divisor M on a normal variety Z is a rule that
assigns a Q-divisor M_{Z'} to every birational model Z'to Z, compatibly
with pushforward. It is b-nef when M_{Z'} is nef on some model and
stays nef under pullback, and b-Cartier when some integer pM_{Z'} is
Cartier on a determining model. Here M denotes the moduli
b-divisor and pbfM b-Cartier means pM_W is an honest Cartier divisor
on a model W that determines M.M.
[(V10)] A generalized pair (Z,B_Z+M_Z) consists of a boundary
B_Z and the trace M_Z of a b-divisor on a model. Its adjoint is
K_Z+B_Z+M_Z, and its discrepancies are read off the crepant boundaries on
higher models. It is generalized lc (or klt) when those discrepancies are
at least -1 (or bigger than -1). The crepant boundaries on higher models
may have negative coefficients; that is part of the definition, not an error.
[(V11)] A DCC set is a subset of R that contains no infinite
strictly decreasing sequence; an ACC set is the reverse. The relevance is
that a DCC set of coefficients cannot contain smaller and smaller rationals
approaching a limit from above, and an ACC theorem says that a collection of
coefficients produced by a geometric process must lie in some DCC set or even a
finite set. The argument uses the ascending chain condition for log canonical
thresholds, quoted from Hacon--McKernan--Xu.

{small./pdf/unknown-nd-1208.4150-small.pdf}
[(V12)] The index of a log Calabi--Yau pair (T,B_T) with
K_T+B_Tsim_{Q}0 is the smallest positive integer n with n(K_T+B_T)sim0.
An index theorem bounds n in terms of the dimension and a finite set of
allowed boundary coefficients. The argument imports the slc index theorem of
Jiang--Liu in dimension at most three.

{small./pdf/unknown-nd-2002.11928-small.pdf}
[(V13)] For an integral Weil divisor G on a normal variety, the
reflexive divisorial sheaf OO_X(G) has sections, on an open U,
the nonzero rational functions v with Div(v)|_U+G|_Uge0. For a rational
divisor D, the rounded system is the complete system of
OO_X(floor( D)): nonzero v with Div(v)+Dge0. This needs no
Cartier condition. The field generated by a system is the subfield of
C(X) generated by ratios of nonzero sections.
[(V14)] A cyclic cover of a smooth curve C at a point c is the
normalization of C in C(C)(w) with w^m a local uniformizer at c. It is
totally ramified of degree m over c and 'etale elsewhere. Pulling back a
fibration to a cyclic cover and comparing a rational generator with its Galois
translates is the mechanism by which a denominator is read off a
character.
[(V15)] The residue of a rational top form with a logarithmic pole
along a smooth divisor is the form obtained by contracting with a local
generator of the normal bundle. Iterated residues give a form on a reduced
normal crossing fibre, and the residues of two generators differ by a scalar
determined by the local node. This is the one analytic-flavoured operation the
paper uses.
[(V16)] Rational connectedness of a smooth projective variety Y
means that any two points are joined by a chain of rational curves. It implies
Pic^0(Y)=0 and, with it, that the N'eron--Severi group is finitely
generated. The divisor class group Cl(Z) of a normal variety with a
rationally connected resolution is then finitely generated, and its torsion
is bounded using Kummer theory.




medskipnoindentIn words. The proof of the main theorem has two halves. The first
half is conceptual: it produces the exact presentation and
turns the moduli bound into a statement about one prime at a time. The second
half, Sections~sec:curve and~sec:residue, is a single hard
computation for that one prime. The whole strategy is: reduce a global claim
to a local number, then compute that number once.

medskipnoindentHalf one, step by step. The argument starts with the generic fibre of f. It
is a log Calabi--Yau pair of dimension d-dim Zle d-1le 3, so the known
index theorem in low dimension applies to it. That theorem provides a degree
p_0 in which the adjoint of the generic fibre is principal, with an explicit
rational function. Because the function field extension is regular, a Galois
descent (Hilbert's theorem 90) moves that function down to the actual generic
fibre, not just its algebraic closure. The result is a function psi on X
whose (1/p_0)-divisor corrects K_X+B to be vertical over Z. A short
argument in the same spirit then chooses D_Z in the rational class of D so
that the corrected adjoint is literally f^*D_Z, with no vertical error left.
That is the exact presentation~(eq:relative-exact-presentation).

Once the presentation is exact, the qualitative lc-trivial fibration theorem
gives the structural facts for free: the discriminant B_Z is
effective and the moduli b-divisor M is b-nef and descends on a
smooth model. What the qualitative theorem does not give for free is a
uniform Cartier index for M.M. That is the content of the argument, and
it is what remains to be reduced.

medskipnoindentTwo independent boundings. The discriminant coefficients 1-t_P
must fall into a fixed DCC set; this comes from ACC for log canonical
thresholds, applied prime by prime, and is purely qualitative and already
uniform. The moduli coefficients must have bounded denominators; this is the
hard part. On a smooth model Wto Z with M_W=D_W-K_W-B_W, a coefficient of
M_W at a prime Psubset W equals an integer (from K_W) plus
coeff_P D_W plus coeff_P(-B_W). If
alpha=coeff_P D_W and B_W has coefficient 1-t_P at P
over the generic point, that coefficient is alpha+t_P, up to an integer.
So it suffices to show that one fixed integer p, independent of the
fibration and of P, has
[
 p(alpha+t_P)inZ
  for every prime Psubset W.
]
This is the target~(eq:relative-coefficient-target) below. Everything in
Sections~sec:curve and~sec:residue is aimed at it.



medskipnoindentTechnical form. Put k=dim Z and let K=C(Z). The contraction
condition makes K algebraically closed in C(X), so the extension
C(X)/K is regular; in characteristic zero it is separably generated, and
the generic fibre X_eta is geometrically integral. It is normal by
localization from X, and normality is geometric for a finite-type variety
over the perfect field K.

Choose a rational top form on Z, and use it together with the form defining
K_X to fix a relative top form on X_eta. Its divisor is the canonical
divisor K_{X_eta}. At codimension-one points of X_eta, which are smooth
over K, this divisor is the coefficientwise localization of K_X, so
vertical primes disappear; when d=k, X_eta=Spec K and the
relative form is a nonzero 0-form. Restricting a log resolution and its
crepant sub-boundary to the geometric generic fibre exhibits the original
geometric generic pair as lc: generic smoothness makes the horizontal strata
simple normal crossing there. That pair is projective of dimension
d-kle 3, its boundary coefficients lie in Phi, and its adjoint is
rationally linearly trivial.

[generic-fibre trivialization]
There is a positive integer p_0, depending only on d and Phi and
clearing the denominators of Phi, and a rational function
psiinC(X)^* such that
[
 p_0(K_X+B)+Div(psi)
]
is vertical over Z.


Proof. [Plain form]
Apply the slc index theorem to the geometric generic fibre in dimension at most
three, getting a trivializing degree. Descend the trivialization across the
Galois action by taking ratios of conjugates, whose divisors are vertical and
hence constant, and clearing the resulting cocycle with Hilbert 90.


Proof. [Technical form]
The geometric generic fibre is a projective lc pair of dimension at most three
with boundary coefficients in Phi and rationally trivial adjoint. The slc
index theorem (Jiang--Liu, quoted as input) gives a common integer p_0,
depending only on d and the finite coefficient set, such that
p_0(K_{X_{overlineeta}}+B|_{X_{overlineeta}}) is principal on the
geometric generic fibre. Choosing p_0 also to clear Phi gives an
integral divisor.

Descend to K. Choose a finite Galois extension L/K over which a
principalizing rational function is defined. The quotient of two Galois
conjugates of it has zero divisor on the proper geometrically integral normal
fibre over L, hence is a nonzero constant. These quotients form a
multiplicative Galois cocycle, and Hilbert's theorem 90 rescales the function
to be Galois-invariant; being defined over K, it descends to C(X)^*.
Thus there is psiinC(X)^* making p_0(K_X+B)+Div(psi) vertical.




medskipnoindentTechnical form. Choose a divisible by p_0 and
phiinC(X)^* with
[
 a(K_X+B)+Div(phi)=f^*(aD).
]
Such a pair exists because K_X+Bsim_{Q}f^*D and a clears denominators.
The divisor of psi^{a/p_0}/phi is vertical. On the proper normal generic
fibre this function has neither zeros nor poles, so it belongs to K^*.
Writing it as h, set
[
 D_Z=D+frac1aDiv(h).
]
Substituting gives the first equality of~(eq:relative-exact-presentation)
as an equality of actual rational divisors. In particular D_Z is rational
Cartier and D_Zsim_{Q}D.



medskipnoindentTechnical form. Apply the lc-trivial-fibration theorem of
Fujino--Gongyo. Besides generic log canonicity and the rational pullback
relation it needs a rank-one condition: on a log resolution the modified
discrepancy divisor drops the discrepancy -1 terms, and because
0le Ble 1 rounding up leaves, over the generic point of the base, only an
effective exceptional divisor; its divisorial sheaf pushes forward to OO_X
by normality, and pushing on to Z has rank one. The theorem then makes
M b-nef and gives its descent to a nef rational Cartier divisor on a
sufficiently high smooth projective model Wto Z. The statement is stable
under changing the rational trivialization, since that adds a rational
principal b-divisor from the base.

Write
[
 D_W=(Wto Z)^*D_Z, M_W=D_W-K_W-B_W.
]
For a prime Psubset W, set alpha=coeff_P D_W. Since
coeff_P B_W=1-t_P and the coefficient of K_W is an integer,
[
 coeff_P M_W
   =alpha+t_P-1-coeff_P K_W.
]
Hence it is enough to prove, uniformly in the choice of P and the fibration,

 p(alpha+t_P)inZ.

Once~(eq:relative-coefficient-target) holds for a single p, the divisor
pM_W is an integral divisor on the smooth variety W, hence Cartier, and
pbfM is b-Cartier.



medskipnoindentTechnical form. For primes Psubset Z, shrink about the generic
point of P so that P is Cartier. In a log resolution computing the
threshold, only divisors with positive order along the pullback of P
matter; discard those whose images are proper subsets of P. Then t_P is an
ordinary log canonical threshold of an lc pair with coefficients in Phi
against an effective Cartier divisor with positive integral coefficients. The
threshold ACC theorem (Hacon--McKernan--Xu) applies, and since thresholds are
rational and lie in [0,1], their complements 1-t_P form the required
rational DCC set B[0,1]. The singularities of
(Z,B_Z+M_Z) are generalized lc (respectively generalized klt) because the
crepant sub-pair over Z is sub-lc (respectively sub-klt): t_Pge0 always,
and t_P>0 in the klt case. On the original base Z the boundary upstairs is
effective with positive integral multiplicity over P, which forces
t_Ple1 and hence B_Zge0.



After these considerations the main theorem is reduced to a single statement: there is
an integer p, depending only on d and Phi, such that the coarse identity

 p(alpha+t_P)inZ
  for every prime P of every determining model W

holds. The threshold t_P is disguised: it was computed as an lc threshold over
the generic point of P, but the required uniformity is insensitive to the
individual P. The next two sections compute~(eq:problem) by slicing to
a curve through the generic point of P, running a relative minimal model
program there, and reading the answer off a residue on the resulting reduced
fibre.



medskipnoindentIn words. The goal is to prove one congruence, p(alpha+t_P)inZ, for
one prime P of one determining model W. The trick is to make the situation
one-dimensional by cutting with general divisors through a general point of
P. A curve has no room for complications: the "fibre" over a point of the
curve is a divisor, the singularities are controlled, and a residue
computation can see the denominator. So slice and run the relative minimal
model program, and reduce the congruence to a statement about a single reduced
special fibre.

medskipnoindentThe plan. Fix Psubset W. Build a smooth model Vto X that maps
to W and is marked along everything relevant. Cut V by k-1=dim W-1
general very ample divisors to get a curve C and a total space V_Cto C.
On V_C write an exact relative identity whose base coefficient at the chosen
point cin C is exactly alpha. Construct a dlt pair by adding the reduced
fibre and a horizontal boundary. Run a minimal model program over C. The
program does not change the function field and does not change the coefficient
of interest; it makes the total space terminal enough that the reduced fibre
is well behaved. The output is a dlt model Nto C on which an exact divisor
relation holds:
[
 L+Div(psi_C)=p_0(alpha+t)f_N^*[c], L=p_0(K_N+T+H),
]
with T the reduced fibre. That relation is handed directly to the residue
lemma of Section~sec:residue.



medskipnoindentTechnical form. Fix a prime Psubset W. Take a smooth projective
model Vto X mapping to W by g: Vto W, with simple normal crossing
markings for the crepant boundary B_V^c, the exceptional divisors over X,
and the support of g^*P. Because the function field extension C(W)subset
C(V) is regular (the same argument as for f), the Stein factorisation of g
is W itself, so g has connected fibres. Let theta_V be the rational top
form compatible with the chosen canonical divisor on X, and put
K_V=Div(theta_V). Crepant pullback gives

 K_V+B_V^c+frac1{p_0}Div(psi)=g^*D_W.

Put t=t_P. At the generic point of P, the boundary B_V^c+t,g^*P has
coefficients at most one, with equality on at least one component dominating
P; this is the threshold computation.

Choose general very ample divisors A_1,...,A_{k-1} on W (taking
C=W when k=1) and let C=A_1cap...cap A_{k-1}. Bertini applied
successively to the pulled-back systems and the finitely many marked strata
makes C and V_C=g^{-1}C smooth, with the markings still SNC. The curve C
is connected; the restriction of g has connected fibres, so V_C is
connected and hence integral. Also required are CnotsubsetSupp D_W and
V_CnotsubsetSupp K_VcupSupp B_V^ccupSuppDiv(psi). The dimension of
V_C is d-k+1le4. Choose an intersection point cin Ccap P transverse and
general on P, avoiding at c all other components of Supp D_W, the bad
strata on P, and the images of all nondominating strata not containing P.



medskipnoindentTechnical form. Let h:V_Cto C, F=h^*[c], and B^c=B_V^c|_{V_C}.
To keep the coefficient of P the canonical divisor is specified using
complete-intersection adjunction. Choose general representatives A_j'sim A_j
avoiding c and rational functions chi_jinC(W)^* with
Div(chi_j)=A_j'-A_j. Define theta_C as the iterated Poincar'e residue,
in the order of the cuts, of
[
 theta_Vprod_{j=1}^{k-1}g^*chi_j
]
along g^{-1}A_1,...,g^{-1}A_{k-1}. Generality makes this a nonzero rational
top form on V_C; for k=1 the product is empty and the residue is the
identity. Each chi_j supplies a simple pole along its divisor and a zero along
its chosen representative, so, cancelling the cutting divisors before
restriction, the residue formula gives an equality of actual divisors
[

 K_{V_C}:=Div(theta_C)
     &=(Div(theta_Vprod_jg^*chi_j)+sum_jg^*A_j)|_{V_C}
     &=(K_V+sum_jg^*A_j')|_{V_C}.

]
Set psi_C=psi|_{V_C} and D_C'=(D_W+sum_jA_j')|_C. Restricting
(eq:relative-resolution-presentation) gives

 K_{V_C}+B^c+frac1{p_0}Div(psi_C)=h^*D'_C,
  coeff_c D'_C=alpha.

The coefficient assertion uses that C meets P transversely at c while the
A_j' and all other components of Supp D_W avoid c.



medskipnoindentTechnical form. Along F, the coefficients of B^c+tF are at most
one, with equality on a component of F: the divisors dominating P restrict
transversely and reducedly, and near c, g^*P|_{V_C}=F. Over the
generic point of C there is still a projective birational comparison with the
normal fibre of the original model X, and the restricted exceptional divisors
are exceptional there; this is because the generic point of C lies where
Wto Z is an isomorphism, and generic flatness and openness of geometric
normality give normal integral original fibres on an open base with an open
birational isomorphism locus whose exceptional images have fibre codimension at
least two.

On V_C define an effective boundary Gamma as F_{red}, plus the
horizontal strict transforms of the original boundary with their original
coefficients, plus all horizontal restricted exceptional divisors with
coefficient one. Then (V_C,Gamma) is log smooth, its horizontal coefficients
lie in the fixed set Phicup{1}, and

 E=Gamma-B^c-tF

is effective near c and has coefficient zero on a component of F. On the
generic fibre, E is effective: its horizontal nonexceptional terms cancel,
while every horizontal exceptional term has coefficient 1-coeff
B^cge0, and its support there is exceptional over the original normal fibre.
Equation~(eq:relative-curve-presentation) gives the exact identity
[
 K_{V_C}+Gamma+frac1{p_0}Div(psi_C)=h^*(D'_C+t[c])+E.
]

Run a minimal model program over C. All negative coefficients of E are
vertical and occur over finitely many points, so there is an effective rational
divisor A on C with E^+:=E+h^*Age0. The identity becomes
[
 K_{V_C}+Gamma+frac1{p_0}Div(psi_C)=h^*(D'_C+t[c]-A)+E^+.
]
In particular K_{V_C}+Gammasim_{Q,C}E^+, whose restriction to a very general
fibre is rationally linearly equivalent to an effective divisor, hence
pseudo-effective over C.



medskipnoindentTechnical form. The existence-and-preservation result of
Chen--Tsakanikas permits a minimal model program with scaling of an ample over
the base for an lc generalized pair whose underlying variety is Q-factorial
klt; applied to the Q-factorial dlt pair (V_C,Gamma) over C, it keeps the
stages Q-factorial and dlt, and each step is negative for the transformed
adjoint. The zero nef datum satisfies the NQC hypothesis. In dimension four,
their theorem terminates every flip sequence over C for an NQC lc generalized
pair whose adjoint is pseudo-effective over C; in dimension three their
statement has no pseudo-effectivity hypothesis. These give the program and
terminate the flip tail.

Fix canonical divisors on every birational stage using theta_C, and push
forward Gamma,E^+; every step is over C, so the identity pushes forward to
[
 K_{V_i}+Gamma_i+frac1{p_0}Div(psi_C)=h_i^*(D'_C+t[c]-A)+E_i^+,
  E_i^+ge0.
]
A fibre-type negative extremal contraction is impossible: a general
positive-dimensional contracted fibre has a curve ell not contained in
Supp E_i^+, and Q-factoriality plus effectivity makes the intersection
E_i^+·ellge0, while the contraction makes it negative, a contradiction.
Divisorial contractions strictly decrease the relative Picard number, so an
infinite program would eventually consist only of flips, which the termination
statements exclude; in dimensions at most two no flips occur. The endpoint is a
projective morphism f_N:Nto C with N Q-factorial, (N,Gamma_N) dlt, and
K_N+Gamma_N nef over C. The unchanged function field and Stein factorisation
give connected fibres, so f_N is a contraction.



medskipnoindentTechnical form. Let E_N^+ be the transform of E^+ on N. The
pushforward of h^*A is f_N^*A, so E_N:=E_N^+-f_N^*A is the pushforward of
the original E, and the transformed identity reads

 K_N+Gamma_N+frac1{p_0}Div(psi_C)=f_N^*(D'_C+t[c])+E_N.

Take a common resolution r:widetilde Vto V_C, r':widetilde Vto N.
Discrepancy comparison for a K+Gamma program gives
[
 r^*(K_{V_C}+Gamma)-r'^*(K_N+Gamma_N)ge0,
]
and cancelling common base pullbacks and principal divisors in the input and
output identities gives

 r'^*E_Nle r^*E.

Let eta be the generic point of C and q:widetilde V_etato X_eta the
map to the normal original fibre. The restriction of E_N to N_eta is
effective, being the pushforward of the effective restriction of E; the
restriction of r^*E is effective and q-exceptional; so by
(eq:relative-error-monotonicity) the restriction of r'^*E_N is
q-exceptional. But (eq:relative-output-error) gives
E_Nequiv_C K_N+Gamma_N, which is nef over C, and hence q-nef. The
negativity lemma makes it zero. Thus E_N has no horizontal component; after
shrinking about c it is effective, supported on the special fibre, and nef
over C.

Write F_N=f_N^*[c]=sum_i e_iS_i and E_N=sum_i q_iS_i with e_i>0,
q_ige0, and set lambda=max_i(q_i/e_i). Then
Delta=lambda F_N-E_N is effective, anti-nef over C, and misses at least one
component of the fibre. If the fibre has positive dimension and Deltane0,
connectedness gives a component S outside SuppDelta meeting it; since N
is Q-factorial a positive multiple bDelta is Cartier, and its restriction
to the integral projective S is a nonzero effective Cartier divisor, so
((bDelta)|_S)· A_S^{n-1}>0 for very ample A_S, contradicting anti-nefness.
Hence Delta=0, i.e. E_N=lambda F_N; for a zero-dimensional fibre, f_N is
an isomorphism of normal curves and the proportionality is immediate. Since
r'^*F_N=r^*F and E has coefficient zero on a component of F, comparing that
coefficient forces lambda=0. So E_N=0 near c.



medskipnoindentTechnical form. Shrink C about c and set
T=(f_N^*[c])_{red}. No divisor was extracted in the program, so every
surviving component of this fibre has coefficient one in Gamma_N; hence
Gamma_N=T+H, with H horizontal and coefficients in Phicup{1}. The
reduced fibre T is connected and projective, and its components have dimension
d-kle3. Removing the boundary from the Q-factorial dlt pair shows that N
is klt. Shrink again so that D'_C is supported at most at c. For the
integral Weil divisor L=p_0(K_N+T+H) the exact relation is now

 L+Div(psi_C)=p_0(alpha+t)f_N^*[c].

This is exactly the hypothesis of the residue lemma, with
beta=alpha+t. The remaining work is to show that one uniform p, depending
only on d and Phi, has pbetainZ in every such configuration.



medskipnoindentIn words. This is the one genuinely delicate computation. The setup provides
handed a normal total space N over a curve, a reduced fibre T, and a
horizontal boundary H, with an exact relation
[
 p_0(K_N+T+H)+Div(psi)=p_0beta,F,
  F=f^*[c], T=F_{red}.
]
The question is whether pbeta is an integer for one fixed p that does not
depend on the configuration. The strategy is: restrict the relation to the
reduced fibre, where it becomes a statement that a certain log Calabi--Yau pair
(T,B_T) has a trivial p-th pluricanonical class; take a global generator u
of that trivialization; pull back to a cyclic cover of the curve that opens up
the denominator; and show, by comparing residues across the nodes of the cover,
that the would-be extra character is forced to be trivial. The denominator must
therefore divide p.

medskipnoindentTechnical form.

[The reduced fibre and the cyclic residue test]
Fix a finite set Phisubset[0,1]capQ and a positive integer p_0 clearing
its denominators. There is a positive integer p, depending only on Phi and
p_0, divisible by p_0, with the following property. Let
[
 f: N→ C
]
be a projective contraction from a normal integral variety to a smooth complex
curve, let cin C, and suppose 1ledim Nle4. Set F=f^*[c] and
T=F_{red}. Assume that N is Q-factorial, that (N,T+H) is dlt,
and that H is an effective horizontal Q-divisor with coefficients in
Phi. Suppose, on a neighbourhood of F, that there are betainQ and
psiinC(N)^* such that, for a fixed canonical divisor,

 p_0(K_N+T+H)+Div(psi)=p_0beta F

as an equality of actual Q-divisors. Then pbetainZ.


Proof. 
The proof has four movements. Adjoin on the whole reduced fibre to get an
slc pair (T,B_T) with a uniform index. Take a global generator u and a
cyclic cover that unwinds the denominator. Compare residues on the components
of the covering fibre and glue the scalars. Kill the remaining character.



medskipnoindentTechnical form. Shrink C about c throughout; fix a uniformizer
zinC(C) with Div_C z=[c] there. The fibre T is connected and
projective, and its irreducible components have dimension dim N-1, because
F is an effective Cartier divisor on the integral variety N with support the
whole fibre. When dim N=1, f is an isomorphism of normal curves and
comparing the coefficient at c already gives p_0betainZ; so assume
1ledim Tle3.

Write T=∪_iT_i. By dlt adjunction and the structure of dlt strata, each
T_i is normal and

 (K_N+T+H)|_{T_i}sim_{Q}K_{T_i}+Θ_i,

where (T_i,Θ_i) is lc and Θ_i is the effective different. If n
is sufficiently divisible, the n-th tensor power of the ordinary residue at
the generic point of T_i extends to an isomorphism
[
 OO_N(n(K_N+T+H))|_{T_i}~=OO_{T_i}(n(K_{T_i}+Θ_i)),
]
identifying actual rational pluricanonical forms.

Intersections of distinct components are unions of lc strata. At the generic
point of a codimension-two intersection in N the pair is simple normal
crossing, exactly two components of T pass through the point, and H is
absent; the conductor prime on either branch has coefficient one in Θ_i.
The remaining coefficients of the differents lie in the set

 D({(r-1+b)/(r) middle|
   rinN_{>0}, b=sum_{phiinPhicup{1}}n_phiphile1}cap[0,1],

which is a fixed rational DCC set: the possible sums ble1 form a finite set,
and for each such sum the displayed coefficients increase in r toward one.

To apply the index theorem to T itself and not just to its components, T must be the right kind of scheme. Apply the depth theorem to the dlt pair
(N,T+H) with the effective integral divisor D=Tlefloor( T+H) and
L=0; it makes OO_N, OO_N(-T), and OO_T Cohen--Macaulay, so T
satisfies S_2. The normality of the T_i and the generic crossing description
make T smooth away from ordinary double curves, hence seminormal (the
square--cube criterion, checked at codimension-one points, plus the S_2
extension property). Thus T is demi-normal, its normalization is the disjoint
union of the T_i, and its conductor is the union of the double curves.

Remove the coefficient-one conductor divisors from the Θ_i and push the
remaining boundary cycles to T; call the result B_T. Off the conductor the
normalization is an isomorphism in codimension one, so nothing is double
counted. The boundary B_T is effective, has coefficients in D(,
and contains no conductor component. Choosing an open j:T^circ↪ T
whose complement has codimension at least two and where T^circ has only
smooth points and ordinary double crossings, one defines
OO_T(n(K_T+B_T)) first on T^circ and then extends by S_2. For every
sufficiently divisible even n the residue maps give a natural isomorphism

 OO_N(n(K_N+T+H))|_T~=OO_T(n(K_T+B_T)).

At a node with normal-crossing coordinates x,y, hypersurface adjunction for
xy=0 matches the ambient log canonical line with the dualizing line of the
node; the two iterated residues of dx/xwedge dy/y differ by a sign whose
n-th power is +1 for even n, and their next residues agree. It follows
that K_T+B_T is Q-Cartier and its pullback to the normalization is
K_{T_i}+Θ_i with the conductor included, so (T,B_T) is slc. Combining
(eq:slc-local-relation) with F=Div(f^*z) and restricting a sufficiently
divisible trivialization in~(eq:slc-ambient-restriction) gives

 K_T+B_Tsim_{Q}0.




medskipnoindentTechnical form. Apply global ACC to the projective normalized
component pairs (T_i,Θ_i), whose adjoints are numerically trivial by
(eq:slc-fibre-trivial). Their dimensions are at most three and their
coefficients lie in the DCC set D(, so their nonzero coefficients
lie in a fixed finite rational set; likewise for B_T. The slc index theorem in
dimension at most three then gives a common integer annihilating K_T+B_T.
Enlarging it, choose once and for all an even multiple p of p_0, depending
only on Phi and p_0, with

 pB_T integral, p(K_T+B_T)sim0.

Choose a nowhere-vanishing generator u of this trivial log pluricanonical line;
on the normalization it is a tuple of rational p-pluricanonical forms u_i
with the prescribed boundary poles and the conductor gluing.



medskipnoindentTechnical form. Write

 p_0beta=(a)/(m), ainZ, minN_{>0}, gcd(a,m)=1.

The case m=1 is immediate; assume m>1. Let C'to C be the normalization
in C(C)(w) with w^m=z, and let pi:Yto N be the normalization of the
dominating component of Ntimes_CC'. The polynomial w^m-z is Eisenstein at
c, so C' has a unique point c' over c with total ramification of degree
m; and since C(C) is algebraically closed in C(N) and the extension is
regular, C(C')/C(C) is linearly disjoint from it. The map pi has degree
m and cyclic Galois group mu_m acting by w|->zeta w. Write
f_Y:Yto C' and T_Y=(pi^{-1}T)_{red}. Figure~fig:root-cover
records the maps; Y is the normalization of the base change.

[ht]
centering
[row sep=large,column sep=huge]
 T_Y arrow[r] arrow[d,hook] & T arrow[d,hook] 
 Y arrow[r,"pi"] arrow[d,"f_Y"'] & N arrow[d,"f"] 
 C' arrow[r,"w^m=z"'] & C

Caption: The normalized cyclic base change and its whole reduced special
fibres over c' and c. The covering group may permute components of
T_Y, so the residue comparison must use their conductor gluing.



The morphism f_Y is projective and its function field is regular over
C(C'), so Stein factorisation gives (f_Y)_*OO_Y=OO_{C'}; in particular
T_Y is connected and projective. The divisor p_0(K_N+T+H) in
(eq:slc-local-relation) is integral, and comparing coefficients with
(eq:slc-denominator) shows m divides every coefficient e_i of
F=sum_ie_iT_i. At the generic point of T_i write f^*z=tau^{e_i}v with
v a unit; after dividing w by tau^{e_i/m} the normalized extension is
obtained by extracting an m-th root of a unit and is 'etale. Away from T
the function f^*z is a unit and the same holds, so pi is 'etale in
codimension one.

Put H_Y=pi^*H and choose canonical divisors compatibly. Since pi is
quasi-'etale and the coefficient-one divisor pulls back without divisorial
ramification,

 K_Y+T_Y+H_Y=pi^*(K_N+T+H).

The pair on the left is dlt. Finite crepant discrepancy comparison multiplies a
downstairs log discrepancy by the ramification index of a divisorial
prolongation, so the upstairs pair is lc and every upstairs lc centre maps onto
a downstairs lc centre; a finite quasi-'etale map is 'etale over the smooth
locus of its target by purity of the branch locus, so the upstairs pair is
simple normal crossing at the generic points of all its lc centres, which is the
dlt condition. In particular each component T_{Y,j} of T_Y is normal.



medskipnoindentTechnical form. Let theta be the rational top form defining
K_N, and define

 s=(pi^*theta)^{ ⊗  p_0},pi^*psi,w^{-a}.

Since pi^*F=mDiv_Yw, equations~(eq:slc-local-relation) and
(eq:slc-cover-crepant) give Div(s)+p_0(T_Y+H_Y)=0. Thus s generates
the degree-p_0 divisorial log pluricanonical sheaf on Y, and its
sufficiently divisible powers generate the corresponding invertible sheaves.



medskipnoindentTechnical form. Let T_i be the component of T dominated by
T_{Y,j}, and let rho_j:T_{Y,j}to T_i be the induced finite 'etale map
at their generic points. The generic residue
[
 v_j=res_{T_{Y,j}}(s^{p/p_0})
]
is a nonzero rational p-pluricanonical form, as is rho_j^*u_i, and the
quotient

 r_j=(v_j)/(rho_j^*u_i)inC(T_{Y,j})^*

is a rational function. Choose a multiple n=bp sufficiently divisible for
(eq:slc-ambient-restriction) and for component adjunction upstairs. Then
v_j^b is a nowhere-vanishing generator of the degree-n adjunction sheaf on
T_{Y,j}. The same is true of rho_j^*u_i^b: near any point of T, the
section u^b is a generator of the restriction of the invertible sheaf
OO_N(n(K_N+T+H)), hence is a unit relative to a local ambient generator; such
a unit lifts after restricting the neighbourhood, so u^b is the residue
restriction of a local ambient log generator. Pull that generator back to Y;
it remains a log generator by (eq:slc-cover-crepant), and its residue on
T_{Y,j} generates the upstairs adjunction sheaf and equals rho_j^*u_i^b
because residue commutes with 'etale pullback. Therefore r_j^b, and hence
r_j, has zero divisor; a rational function with zero divisor on a normal
projective integral variety is a nonzero constant. So

 v_j=r_j,rho_j^*u_i, r_jinC^*.




medskipnoindentTechnical form. Suppose two distinct components T_{Y,j} and
T_{Y,j'} meet. Their intersection contains a stratum of codimension two in Y
along which the pair is generically simple normal crossing, and its image is a
codimension-two lc centre downstairs where pi is 'etale and the downstairs
pair is simple normal crossing. The two downstairs vertical branches are
distinct, and H is absent at this generic crossing. At the generic point xi
of the upstairs crossing choose the two normal coordinates x,y and a regular
ambient (dim Y-2)-form eta. Since s^{p/p_0} is a log generator, it has
the local form
[
 s^{p/p_0}=ε((dx)/(x)wedge(dy)/(y)wedgeeta)^{ ⊗  p},
  εinOO_{Y,xi}^*.
]
The two next residues differ by (-1)^p and are nonzero, because ε
is a unit; as p is even they agree. The components of u have the same
matching, nonzero next residues downstairs, since u generates the slc log
pluricanonical line whose local node description is the even tensor power of the
dualizing line, and 'etale pullback preserves this. Taking next residues in
(eq:slc-component-scalar) gives r_j=r_{j'}.



medskipnoindentTechnical form. Since T_Y is connected, its component-incidence
graph is connected, so all r_j equal one scalar rinC^*. In tuple notation
on the normalized components of T_Y,

 (res_{T_{Y,j}}(s^{p/p_0}))_j=r,pi^*u,

and pi^*u is invariant under the covering group, with the natural action also
permuting components. On the other hand, for zetainmu_m,
(eq:slc-cover-generator) gives zeta^*(s^{p/p_0})=zeta^{-ap/p_0}s^{p/p_0},
and residue is natural under this action including its permutation of
components. The nonzero scalar comparison~(eq:slc-global-scalar) forces
this character to be trivial. So mmid ap/p_0; since gcd(a,m)=1, obtain
mmid p/p_0, and hence
[
 pbeta=(p)/(p_0)·(a)/(m)inZ.
]
This proves the lemma.




Apply Lemma~lem:slc-residue-test to (eq:relative-local-cyclic) with
beta=alpha+t and the fixed coefficient set Phicup{1}. It gives a single
multiple p of p_0, depending only on d and Phi, for which
(eq:relative-coefficient-target) holds at every prime P of the
determination W. Consequently pM_W is integral and Cartier, and since the
moduli b-divisor descends on W the b-divisor pbfM is b-Cartier. Every
construction used only the dimension bound dle4 and the fixed coefficient
set Phi. Together with the discriminant bound of Section~sec:generic
this completes the proof of Theorem~thm:relative-denominators.



medskipnoindentIn words. The main theorem was about writing the base divisor with
bounded denominators. A closely related question is: can pluricanonical
sections upstairs actually see the base downstairs? When the base
divisor is big, the answer is yes in one uniform degree: for every large enough
multiple l, the complete rounded system |floor( l(K_X+B))| is
nonempty, and the field generated by ratios of its sections is exactly the base
function field C(Z), no more and no less. ``No more'' is the substantive
half: the sections must not accidentally generate a larger subfield of
C(X).

medskipnoindentTechnical form.

[Effective systems recovering the full base field]
Under the hypotheses of Theorem~thm:relative-denominators, suppose in
addition that D is big. There is a positive integer m=m(d,Phi) such that,
for every positive multiple l of m, the complete divisorial system
[
 |leftfloor( l(K_X+B)right)|
]
is nonempty and generates precisely the subfield C(Z)subsetC(X). Its
rational map is therefore birationally equivalent to f and to the Iitaka
fibration of K_X+B.


Proof. [Plain form]
First identify the complete section space upstairs with the pullback of a
complete section space on the base, using the exact presentation and a valuation
argument valid for reflexive sheaves. Then produce birational sections on the
base by the polarized effective-birationality theorem applied to a big adjoint
on a smooth model. To identify the Iitaka fibration, observe that multiples of
m form a cofinal divisible subsequence of the Cartier section series, which
has the same field of ratios.


Proof. [Technical form]
Put L=K_X+B and K=C(Z), and use the exact
presentation~(eq:relative-exact-presentation). For every positive integer
l divisible by p_0, the equality of subspaces of C(X) is first proved

 H^0!(X,OO_X(floor( lL)))
   =psi^{l/p_0}f^*
      H^0!(Z,OO_Z(floor( lD_Z))),

where f^* on the right means pullback of rational functions. For
vin K^*, the exact divisor identity gives

 lL+Div_X!(psi^{l/p_0}f^*v)
    =f^*!(lD_Z+Div_Z(v)).

If v is a section of the rounded system on Z, then
lD_Z+Div_Z(v)ge0 by the section convention of Section~sec:vocab. This
divisor is rational Cartier, so local effective Cartier multiples show that its
pullback is effective; (eq:relative-section-divisors) gives the inclusion
from right to left in~(eq:relative-section-spaces).

Conversely, let une0 be a section on the left and let X_eta be the
generic fibre over K. The exact presentation rewrites the section ineuality
as Div_X(u/psi^{l/p_0})+lf^*D_Zge0, whose horizontal valuations give
[
 Div_{X_eta}!(u/psi^{l/p_0})ge0.
]
The generic fibre is normal, so this quotient is regular there; and the
contraction condition f_*OO_X=OO_Z gives
H^0(X_eta,OO_{X_eta})=K. Hence u=psi^{l/p_0}f^*v for some vin K^*. By
(eq:relative-section-divisors) the source ineuality is f^*Gge0 where
G=lD_Z+Div_Z(v) is rational Cartier. Effectivity of G descends to Z:
near the generic point of any prime Psubset Z, write a Cartier multiple of
G as aP with P locally principal; the local Cartier pullback of P has a
prime component dominating P with positive multiplicity e because f is
surjective, and its coefficient in the pullback of the chosen multiple is ae.
Thus f^*Gge0 forces age0. Doing this for every P proves Gge0, so v
is a section on Z, and (eq:relative-section-spaces) holds. Since p_0
clears Phi, the divisor lL is integral Weil, and the argument applies to its
reflexive sheaf even when lL is not Cartier.

Now produce birational sections on the base. Choose a smooth projective
determination q:Wto Z of the nef data that also resolves the boundary, and
write
[
 K_W+B_W^{cr}+M_W=q^*D_Z.
]
Let A be the strict transform of B_Z plus the reduced exceptional divisor.
On nonexceptional primes A agrees with B_W^{cr}; on exceptional
primes its coefficient is one, at least the coefficient of B_W^{cr} by
generalized log canonicity. Thus

 K_W+A+M_W=q^*D_Z+E_W, E_W=A-B_W^{cr}ge0
        exceptional over Z.

Put Λ=B{1}. The pair (W,A) is an ordinary log smooth lc
pair with coefficients in the fixed DCC set Λ, and pM_W is nef Cartier.
Since D is big and D_Zsim_{Q}D, the adjoint on the left of
(eq:relative-big-model) is big.

For each 1le kle d let b_k=b(Λ,k,p) be the integer in the
polarized effective-birationality theorem of Birkar--Zhang. Its conclusion is
that |floor( l(K_W+A+M_W))| is birational for every positive l
divisible by b_k when dim W=k, the floor taken on the whole adjoint. Set
[
 m=lcm(p_0,b_1,...,b_d),
]
an integer depending only on d and Phi. Fix a positive multiple l of m
and take k=dim Z. For a rational section vinC(W)=K of the displayed
birational system, pushing its divisor ineuality forward using
(eq:relative-big-model) gives Div_Z(v)+lD_Zge0. Thus these sections
belong to the complete rounded system on Z, and their ratios generate K
because the system on W is birational. The complete system on Z is therefore
nonempty and generates K, and (eq:relative-section-spaces) transfers
exactly these ratios to X: the common factor psi^{l/p_0} cancels. The
complete source system is nonempty and generates the embedded field Ksubset
C(X) for every such l.

{small./pdf/unknown-nd-1411.2118-small.pdf}

The function field of the image of a system's rational map is generated by its
section ratios, so each of these maps is birationally equivalent to f. To
compare with the Iitaka fibration, choose for the individual pair an integer
r>0 such that rL is Cartier. Degrees divisible by lcm(m,r)
form a cofinal divisible subsequence of the Cartier section series of L, and
such a subsequence has the same field of section ratios: given u/v in one
degree, choose age1 with the multiplied degree in the subsequence and write
u/v=(uv^{a-1})/v^a. The field defining the Iitaka fibration is therefore K,
and kappa(L)=trdeg_{C}K=dim Z. This identifies f with the Iitaka
fibration up to birational equivalence. The individual index r is used only
for this comparison and does not enter m.




medskipnoindentIn words. The moduli denominator theorem says the adjoint on the
base can be written with bounded denominators. When the base is reasonably
simple, one can do better: a uniform multiple of the adjoint is not merely
integral, it is the divisor of an actual rational function. ``Reasonably
simple'' means rationally connected, the algebraic-geometric analogue of
connected-by-rational-curves, which forces the linear-algebraic obstruction (the
Picard group, and its torsion) to be bounded. The plan is: bound the boundary
coefficients and the singularities by global ACC, bound the torsion of the
divisor class group by boundedness, then clear denominators and apply the
torsion bound.

medskipnoindentTechnical form.

[Uniform principal multiple over a rationally connected base]
Fix an integer 1le dle4, a DCC set Isubset[0,1]capQ, and a positive
integer p. There exists a positive integer ell=ell(d,I,p) with the
following property. Let Z be a normal projective variety of dimension d
with a rationally connected smooth projective resolution. Suppose
(Z,B+M_Z) is generalized klt, Bge0 has coefficients in I, and the
b-nef rational b-divisor M has pbfM b-Cartier. If
D=K_Z+B+M_Zsim_{Q}0, then ell D is an integral principal divisor.
Consequently aell D is principal and h^0(Z,OO_Z(aell D))=1 for every
integer age1.


Proof. 
There are three steps: global ACC gives a finite coefficient set and a uniform
positive lower bound for generalized log discrepancies; boundedness up to
isomorphism in codimension one bounds torsion in the divisor class group; and
clearing the coefficients of the actual adjoint applies that torsion bound.



medskipnoindentTechnical form. Suppose an exceptional prime E over Z is marked
with generalized log discrepancy ain(0,1). Choose a high projective log
resolution g:Wto Z on which the nef data descend and E appears, and write
K_W+B_W+M_W=g^*D with Supp B_W SNC and all coefficients less than one.
Arrange also for an effective exceptional Cartier divisor F on W with -F
g-ample and Supp(B_W)cupSupp F SNC, by resolving the singular locus and
then the marked valuation with centres of codimension at least two so that the
negative of each new exceptional divisor is relatively ample.

Choose a small positive rational delta with all coefficients of
B_W+delta F still below one and, when E is marked,
deltamult_E F<a. For a positive ample Cartier A on Z, the divisor
M_W+g^*A-delta F is ample; a general member Λ of a divisible multiple
makes (W,B_W+delta F+Λ) sub-klt. There is a principal P on W with
delta F+Λ=M_W+g^*A+P, and pushing forward gives K_Z+B+g_*Λ=
D+A+g_*P with g^*(K_Z+B+g_*Λ)=K_W+B_W+delta F+Λ. So, with
Xi=B+g_*Λ, the pair (Z,Xi) is an ordinary effective rational klt
pair, and a marked E has ordinary log discrepancy a-deltamult_E Fin(0,1).

Apply the extraction corollary of BCHM to the klt pair (Z,Xi) with the
prescribed set {E} or the empty set. This gives a projective birational
h:Z^+to Z with Z^+ Q-factorial and with exactly the prescribed
exceptional primes (a small Q-factorial modification in the empty case).
Restore the generalized data by K_{Z^+}+B^++M_{Z^+}=h^*D with the same nef
b-divisor; the pair is generalized klt with rationally trivial adjoint, and
B^+ is effective, being the strict transform of B together with (1-a)E in
the extraction case.

Now apply global ACC. If the nef trace is not numerically trivial on a model
determining M,M, write M_W=frac1p(pM_W) with pM_W nef Cartier, and
use the generalized global ACC theorem: the nef weight 1/p is a single fixed
weight adjoined to the DCC set, so the boundary coefficients lie in a finite
set. If instead M_Wequiv0, take a common model widetilde Wto Z^+; the
trace M_{Z^+} is Q-Cartier and the difference M_{widetilde W}-pi^*M_{Z^+}
is exceptional and numerically trivial over Z^+, so the negativity lemma makes
it zero, hence M_{Z^+}equiv0 and the generalized discrepancies are the ordinary
discrepancies of (Z^+,B^+); ordinary global ACC again gives finitely many
coefficients. Applying this first to the small modification with no marked
divisor shows the coefficients of B lie in a fixed finite set I_0subset I
and the pairs Z are uniformly generalized ε-lc for some
ε=ε(d,I,p)>0: otherwise a sequence of marked valuations
with discrepancies decreasing to zero would, after extraction, produce boundary
coefficients 1-a_i in a DCC set that global ACC would force to be finite, a
contradiction.



medskipnoindentTechnical form. Birkar's boundedness theorem applies to the
varieties Z: projectivity, fixed dimension, generalized ε-lc
singularities, effective boundary, rationally trivial adjoint, and rational
connectedness. It gives, for each Z, a projective V_Z, from one bounded set,
isomorphic to Z in codimension one; replacing by finite normalization keeps
this and stays bounded. Hence a projective morphism V S over a
finite-type base whose normal fibres are isomorphic to the Z in codimension
one.

Let U=Z_{reg}. First, H_1(U(C),Z) is invariant under isomorphism
in codimension one of normal varieties: the common open subset, intersected with
the regular loci, is each regular locus with a closed subset of complex
codimension at least two removed, and removing such a subset from a smooth
complex variety does not change the fundamental group (perturb paths and
homotopies, relative to boundary, transverse to a smooth stratification whose
strata have real codimension at least four). Second, the groups take only
finitely many types. Replace S by its reduction and cover by finitely many
flat affine strata with U=Sm( V/S)V/S) the
relative smooth locus; embed V{C}times S and identify
PP^N(C) with Hermitian matrices H^2=H, tr H=1. The map
U( S(C) is continuous semialgebraic, so the Delfs--Knebusch
semialgebraic triviality theorem partitions S(C) into finitely many pieces
over which the whole inverse image is a product; each semialgebraic fibre has the
homotopy type of a finite complex. Thus the H_1 take only finitely many
finitely generated isomorphism types.

For each Z, Cl(Z) is finitely generated: a smooth projective rationally
connected resolution r:Yto Z gives a surjection Pic(Y)toCl(Z) (strict
transforms), and since the Albanese map of Y is constant,
Pic^0(Y)=0, so Pic(Y) is finitely generated by N'eron--Severi finiteness.
Normality and projectivity give Gamma(U,OO_U^*)=C^*, and
Pic(U)=Cl(Z). For every nge1 the 'etale Kummer sequence and
classification of finite abelian torsors give

 Cl(Z)[n]~= H^1_{'et}(U,mu_n)~=
 Hom(H_1(U(C),Z),mu_n(C)).

For a fixed Z the orders on the left are bounded because Cl(Z) is finitely
generated; if H_1(U(C),Z) had a free summand of positive rank the orders on
the right would grow without bound, so it is finite. Finitely many finite groups
occur, so there is a uniform T divisible by all their exponents, and every
torsion class of Cl(Z) is killed by T: take n its order in
(eq:rc-kummer), whose right side has exponent dividing T.



medskipnoindentTechnical form. Since pM_Z is integral Weil, being the pushforward
of a Cartier divisor on a determining model, choose q divisible by p and by
all denominators in the finite set I_0; then qD is integral Weil. The
relation Dsim_{Q}0 makes [qD]inCl(Z) torsion: if nD=Div(v) then
n(qD)=Div(v^q). Hence TqD is principal, and ell=Tq proves the
proposition in dimension d; a common multiple over the finite dimensional range
handles the stated subset. All positive multiples of ell D are principal, and
projectivity with integrality gives the one-dimensional section space.


medskipnoindentFinally, the corollary pulling the two theorems together.

[Uniform principal multiple from a fibration]
Fix a finite rational coefficient set Phisubset[0,1]capQ. There is a
positive integer r=r(Phi) such that the following holds. Let (X,B) be a
projective klt fourfold pair with effective boundary having coefficients in
Phi, and suppose K_X+Bsim_{Q}0. If there is a contraction f:Xto Z with
dim Z>0 and with a rationally connected smooth projective resolution of Z,
then r(K_X+B) is an integral principal divisor. This includes B=0, and every
positive multiple is also principal.


Proof. [Plain form]
Apply the uniform denominator theorem with zero base class. The base adjoint is
rationally trivial, so the rationally connected torsion proposition applies to
it, giving a principal multiple. Multiplying the exact presentation by that
multiple writes the total multiple as the divisor of an actual function.


Proof. [Technical form]
Apply Theorem~thm:relative-denominators to f with the zero divisor as
the rational pullback class. It gives uniformly bounded integers p_0mid p, a
function psiinC(X)^*, and an exact identity

 K_X+B+frac1{p_0}Div(psi)=f^*D_Z, D_Z=K_Z+B_Z+M_Z,

with (Z,B_Z+M_Z) generalized klt, B_Z effective with coefficients in a fixed
rational DCC set, and pbfM b-Cartier with nef data. Since
f^*D_Zsim_{Q}0 and f_*OO_X=OO_Z, the projection formula makes
D_Zsim_{Q}0.

Proposition~prop:rc-torsion gives a uniform ell and vinC(Z)^* with
ell D_Z=Div(v). Choose r divisible by both ell and p, so p_0mid r.
Multiplying (eq:rc-exact-relative) by r gives
[
 r(K_X+B)=Div!((vcirc f)^{r/ell},psi^{-r/p_0}).
]
Both exponents are integers, so this is the divisor of an actual rational
function and includes the integrality of the adjoint multiple. All constants
depend only on Phi, since the possible dimensions of Z form a finite set and
the denominator theorem supplies uniform p_0,p and a fixed DCC set.


appendix



The following table collects symbols used in more than one place. A section
reference points to the introduction of the symbol.

small
{@{}p{.16textwidth}p{.52textwidth}p{.22textwidth}@{}}

Symbol & Meaning & Section 

(X,B) & projective lc pair of dimension dle4, boundary B & sec:question 
Phi & fixed finite set of allowed boundary coefficients in [0,1]capQ & sec:question 
f:Xto Z & contraction with log Calabi--Yau adjoint & sec:question 
K_X+Bsim_{Q}f^*D & rational pullback hypothesis & sec:question 
B_Z & discriminant divisor, coefficient 1-t_P & sec:question 
t_P & log canonical threshold of f^*P over the generic point & sec:question 
M,M, M_W & moduli b-divisor and its trace on a determining model W & sec:generic 
p_0 & generic-fibre trivializing degree, clears Phi & sec:generic 
p & uniform multiple making M b-Cartier & sec:generic 
psi & principal correction with vertical divisor & sec:generic 
D_Z & exact base representative, D_Z=K_Z+B_Z+M_Z & sec:generic 
B & DCC set of discriminant coefficients & sec:generic 
alpha & coefficient of D_W at a prime Psubset W & sec:generic 
V_Cto C & curve slice of a smooth model & sec:curve 
F=h^*[c] & special fibre over the sliced point c & sec:curve 
Gamma, E & dlt boundary and the error Gamma-B^c-tF & sec:curve 
Nto C & endpoint of the relative minimal model program & sec:curve 
T=F_{red} & reduced special fibre, connected & sec:residue 
H & effective horizontal boundary with coefficients in Phi & sec:residue 
(T,B_T) & slc log Calabi--Yau pair obtained by adjunction on T & sec:residue 
D( & DCC set of different coefficients & sec:residue 
a/m & p_0beta in lowest terms, the denominator to kill & sec:residue 
C'to C & cyclic cover w^m=z & sec:residue 
pi:Yto N & normalized base change, Galois group mu_m & sec:residue 
s & ambient generator (pi^*theta)^{ ⊗  p_0}pi^*psi,w^{-a} & sec:residue 
u & global generator of p(K_T+B_T) & sec:residue 
r_j & residue ratio on a covering component & sec:residue 
m=m(d,Phi) & degree generating C(Z) from sections & sec:effective 
ell=ell(d,I,p) & uniform principal multiple on the base & sec:torsion 
r=r(Phi) & uniform principal multiple in the fourfold corollary & sec:torsion 


normalsize

medskipnoindentConventions. The symbols sim, sim_{Q}, and equiv denote linear,
rational linear, and numerical equivalence. Tensor products of line bundles are
written additively: A+B means A ⊗  B and mA means A^{ ⊗  m}.
Equalities such as~(eq:relative-exact-presentation) are equalities of
actual rational divisors for compatible choices of canonical divisors and the
displayed rational function; the canonical divisors are fixed using rational top
forms and are not rescaled silently. An integral principal multiple rD means
rD=Div(v) for an actual rational function v, which includes integrality of
every coefficient of rD. The rounded system of a rational divisor D is the
complete system of OO_X(floor( D)); a nonzero v is a section exactly
when Div(v)+Dge0, and this needs no Cartier condition. The field generated by
a system is the subfield of C(X) generated by ratios of its nonzero sections.
The two integers p_0 and p satisfy p_0mid p; all bounds depend only on the
dimension dle4 and the coefficient set Phi, never on the individual
fibration.