==== TABLE OF CONTENTS ====
Maximal-Multipoint-Seshadri-Constants-in-Higher-Dimensions//compression.tex
Maximal-Multipoint-Seshadri-Constants-in-Higher-Dimensions//conclusion.tex
Maximal-Multipoint-Seshadri-Constants-in-Higher-Dimensions//flag.tex
Maximal-Multipoint-Seshadri-Constants-in-Higher-Dimensions//introduction.tex
Maximal-Multipoint-Seshadri-Constants-in-Higher-Dimensions//jets.tex
Maximal-Multipoint-Seshadri-Constants-in-Higher-Dimensions//lattice.tex
Maximal-Multipoint-Seshadri-Constants-in-Higher-Dimensions//main.tex
Maximal-Multipoint-Seshadri-Constants-in-Higher-Dimensions//nef.tex
Maximal-Multipoint-Seshadri-Constants-in-Higher-Dimensions//node.tex
Maximal-Multipoint-Seshadri-Constants-in-Higher-Dimensions//preamble.tex
Maximal-Multipoint-Seshadri-Constants-in-Higher-Dimensions//provenance/companion/main.tex
Maximal-Multipoint-Seshadri-Constants-in-Higher-Dimensions//provenance/companion/sections/compression.tex
Maximal-Multipoint-Seshadri-Constants-in-Higher-Dimensions//reshaping.tex
Maximal-Multipoint-Seshadri-Constants-in-Higher-Dimensions//saturation.tex
Maximal-Multipoint-Seshadri-Constants-in-Higher-Dimensions//sections/introduction.tex
Maximal-Multipoint-Seshadri-Constants-in-Higher-Dimensions//sections/reshaping.tex
Maximal-Multipoint-Seshadri-Constants-in-Higher-Dimensions//transfers.tex
Maximal-Multipoint-Seshadri-Constants-in-Higher-Dimensions//triangle.tex
Maximal-Multipoint-Seshadri-Constants-in-Higher-Dimensions//provenance/companion/references.bib
Maximal-Multipoint-Seshadri-Constants-in-Higher-Dimensions//references.bib
===========================

==== BEGIN Maximal-Multipoint-Seshadri-Constants-in-Higher-Dimensions//compression.tex ====
\begin{figure}[tbp]
\centering
\begin{tikzpicture}[x=.86cm,y=.90cm,font=\small,
  line cap=round,line join=round,>=stealth]
  \definecolor{diagramblue}{RGB}{40,88,125}
  \definecolor{diagramorange}{RGB}{161,80,28}
  \begin{scope}
    \fill[diagramblue!9] (0,0)--(3.2,0)--(0,2.4)--(-1.6,1.6)--cycle;
    \draw[->,gray!70] (-2.0,0)--(3.65,0) node[right,text=black] {$q$};
    \draw[->,gray!70] (0,-.12)--(0,3.1) node[above,text=black] {$\beta$};
    \draw[diagramblue,thick] (0,0)--(3.2,0)--(0,2.4)--(-1.6,1.6)--cycle;
    \draw[densely dashed,gray!65] (-.8,0)--(-.8,.8);
    \draw[diagramorange,line width=2pt] (-.8,.8)--(-.8,2.0);
    \draw[diagramorange] (-.93,.8)--(-.67,.8) (-.93,2.0)--(-.67,2.0);
    \node[diagramorange] at (-2.0,.62) {$\ell_t(q)$};
    \draw[diagramorange,thin] (-1.62,.78)--(-.94,1.34);
    \node[anchor=west,fill=white,inner sep=1.3pt] at (-.55,.76) {$-q$};
    \fill (0,0) circle (1.2pt) (3.2,0) circle (1.2pt)
          (0,2.4) circle (1.2pt) (-1.6,1.6) circle (1.2pt);
    \node[below right,inner sep=3pt] at (0,0) {$(0,0)$};
    \node[below,inner sep=3pt] at (3.2,0) {$(a,0)$};
    \node[above left,inner sep=3pt] at (0,2.4) {$(0,c)$};
    \node[above left,inner sep=3pt] at (-1.6,1.6) {$(-b,b)$};
    \node[below,inner sep=3pt] at (-.8,0) {$q<0$};
    \node[align=center] at (.7,-.88) {$Q_t$ in $(q,\beta)$ coordinates};
  \end{scope}
  \draw[->,thick] (4.0,1.25)--(5.0,1.25);
  \begin{scope}[shift={(7.4,0)}]
    \fill[diagramblue!9] (-1.6,0)--(3.2,0)--(0,2.4)--cycle;
    \draw[->,gray!70] (-2.0,0)--(3.65,0) node[right,text=black] {$q$};
    \draw[->,gray!70] (0,-.12)--(0,3.1) node[above,text=black] {$h$};
    \draw[diagramblue,thick] (-1.6,0)--(3.2,0)--(0,2.4)--cycle;
    \draw[diagramorange,line width=2pt] (-.8,0)--(-.8,1.2);
    \draw[diagramorange] (-.93,0)--(-.67,0) (-.93,1.2)--(-.67,1.2);
    \node[diagramorange] at (-2.0,.83) {$\ell_t(q)$};
    \draw[diagramorange,thin] (-1.57,.83)--(-.94,.64);
    \fill (-1.6,0) circle (1.2pt) (3.2,0) circle (1.2pt)
          (0,2.4) circle (1.2pt);
    \node[below,inner sep=3pt] at (-1.6,0) {$(-b,0)$};
    \node[below,inner sep=3pt] at (3.2,0) {$(a,0)$};
    \node[above left,inner sep=3pt] at (0,2.4) {$(0,c)$};
    \node[align=center] at (.7,-.88) {$P_t$: each slice starts at height zero};
  \end{scope}
\end{tikzpicture}
\caption{Diagonal compression of exponent slices. For $c(a+b)>ab$, the
change $q=\alpha-\beta$ turns $Q_t$ into the quadrilateral on the left.
Moving each vertical slice down to height zero gives $P_t$, with the
same slice length $\ell_t(q)$. The highlighted slice has $q<0$ and lower
endpoint $-q$. This diagram describes the containing regions; the jet
transfer is proved by convergence of finite jet matrices.}
\label{fig:compression}
\end{figure}
==== END Maximal-Multipoint-Seshadri-Constants-in-Higher-Dimensions//compression.tex ====

==== BEGIN Maximal-Multipoint-Seshadri-Constants-in-Higher-Dimensions//conclusion.tex ====
\section{Proof of the main theorem}
\label{sec:conclusion}

We now assemble the exponent construction and interpolation results. The
reshaping parameters will depend on the point count $r$, but a single
threshold will allow every sufficiently large integer $r$. The final
passage from jets to intersection numbers uses ordinary curve
multiplicities, including at singular points of the curves.

\subsection{The exponent sets for every sufficiently large point count}

Fix the polarized variety $(X,L)$ of Theorem~\ref{thm:main}, and write
$V=L^n$. Choose an integer $D>0$ for which $DL$ is very ample.
Proposition~\ref{prop:flag-simplex} supplies additive support sets $S_k$ with
\[
 S_k\subset kP_0\cap\Z^n,
 \qquad
 \#S_k=h^0(X,kL).
\]
Here $P_0=\Delta(1/D,\ldots,1/D,D^{n-1}V)$.
Let $\ell>0$ be the smallest intercept of $P_0$. Choose an integer $r_0$
such that
\begin{equation}
 \label{eq:threshold}
 (V/r)^{1/n}<\ell/100\qquad(r\ge r_0).
\end{equation}
Such a choice exists because $X$ and $L$ are fixed.

Fix any integer $r\ge r_0$, and put $w=(V/r)^{1/n}$. Apply
Proposition~\ref{prop:reshape} to the first two intercepts, replacing them
by $w$ and their product divided by $w$. Keep the intercept $w$ fixed, and
pair the other new intercept with the next untouched one. Continue until
$n-1$ intercepts equal $w$.

Every pair of input intercepts $A,B$ in this procedure satisfies
$A,B\ge\ell$. Indeed, an untouched intercept has this property, and the
new carried intercept $AB/w$ is greater than $A$, since $B\ge\ell>w$.
Thus the hypothesis of Proposition~\ref{prop:reshape} holds at every step:
\[
 w\max(3/A,2/B)\le\frac{3w}{\ell}<\frac1{16}.
\]
The product of all intercepts remains $V$. We obtain finite sets $F_k$ in
\begin{equation}
 \label{eq:final-simplex}
 kP_*,\qquad
 P_* =\Delta(w,\ldots,w,V/w^{n-1})
     =\Delta(w,\ldots,w,rw),
\end{equation}
with
\[
 F_k+F_l\subset F_{k+l},
 \qquad
 \#F_k=h^0(X,kL)=\frac{V}{n!}k^n+o(k^n).
\]
Since $\vol(P_*)=V/n!$, these sets satisfy the hypotheses of
Proposition~\ref{prop:asymptotic-jets}.

For each $0<\theta<w$ and all sufficiently large $k$, that proposition
gives simultaneous jets through degree $\lfloor k\theta\rfloor$ for
$\M(F_k)$ at an $r$-tuple of distinct torus points. Each application of
Proposition~\ref{prop:reshape} transfers this surjectivity to the preceding
support set. Proposition~\ref{prop:flag-simplex} finally transfers it to
$H^0(X,kL)$ at some tuple of distinct points of $X$. In particular, for
every such $\theta$ and $k$, the set
\begin{equation}
 \label{eq:surjectivity-locus}
 \left\{\mathbf p\in U_r(X):
 H^0(X,kL)\longrightarrow
 \bigoplus_{i=1}^r\Jet_{p_i}^{\lfloor k\theta\rfloor}(kL)
 \text{ is surjective}\right\}
\end{equation}
is nonempty. The tuple witnessing this assertion may depend on both
$\theta$ and $k$.

\subsection{A common very-general locus}

For fixed integers $k\ge1$ and $m\ge0$, the locus of tuples at which
$H^0(X,kL)$ surjects onto jets through degree $m$ is Zariski open in
$U_r(X)$. Indeed, the bundle of principal parts of order $m$ of $kL$ is
locally free because $X$ is smooth. Pulling it back along each coordinate
projection of $U_r(X)$ gives an algebraic vector-bundle map
\[
 H^0(X,kL)\otimes\mathcal O_{U_r(X)}
 \longrightarrow
 \bigoplus_{i=1}^r\operatorname{pr}_i^*\mathcal P^m(kL).
\]
Its surjectivity locus is defined by the nonvanishing of maximal minors.

Choose $0<\theta_j<w$ with $\theta_j\uparrow w$. For each $j$, choose
$k_j$ so that the locus in \eqref{eq:surjectivity-locus} is nonempty and
$\lfloor k\theta_j\rfloor\ge1$ for every $k\ge k_j$. Denote this open
locus by $O_{j,k}$ and set
\begin{equation}
 \label{eq:exceptional-set}
 Z_r=\bigcup_{j\ge1}\ \bigcup_{k\ge k_j}
       \bigl(U_r(X)\setminus O_{j,k}\bigr).
\end{equation}
Every set in this countable union is a proper Zariski-closed subset of
$U_r(X)$.

The complement is nonempty. The complex points of $U_r(X)$ form a
nonempty locally compact Hausdorff space, and hence a Baire space. Since
$U_r(X)$ is smooth and irreducible, every proper algebraic closed subset
has empty interior in its complex topology. A countable union of these
closed subsets cannot cover $U_r(X)$. Thus every tuple outside $Z_r$
simultaneously has all the jet-surjectivity properties indexed by $j$ and
$k\ge k_j$.

\subsection{From jets to ordinary curve multiplicities}

The relation between Seshadri constants and jet separation appears in
Demailly's one-point formulation \cite[Section~6]{Demailly1992} and in the
weighted multipoint framework of \citet[Section~5]{Ito2013}. The following elementary
implication is the one needed here. We give its local proof, including at
singular points of the curves.

\begin{lemma}[Jets bound curve intersections]
\label{lem:jets-curves}
Let $X$ be a smooth complex projective variety, let $A$ be a line bundle
on $X$, and let $p_1,\ldots,p_r$ be distinct points. Suppose that, for an
integer $m\ge1$, the map
\[
 H^0(X,A)\longrightarrow\bigoplus_{i=1}^r\Jet_{p_i}^{m}(A)
\]
is surjective. Then every integral curve $C\subset X$ satisfies
\[
 A\cdot C\ge m\sum_{i=1}^r\mult_{p_i}C
\]
provided that $C$ meets at least one of the points.
\end{lemma}

\begin{proof}
Choose a point $p_i\in C$. The local ring $R=\mathcal O_{C,p_i}$ is a
one-dimensional Noetherian local domain. Its maximal ideal $\frakm_R$
satisfies
\[
 \frakm_R^m/\frakm_R^{m+1}\ne0.
\]
Otherwise Nakayama's lemma would give $\frakm_R^m=0$, contrary to
$\dim R=1$. The quotient map from the ambient local ring to $R$ lifts a
nonzero element of this quotient to a jet on $X$ with no terms of degree
less than $m$. Prescribe this jet at $p_i$ and the zero jet at every other
marked point. By surjectivity, a section $s\in H^0(X,A)$ has these jets.
It vanishes to order at least $m$ at every marked point, and its restriction
to $C$ is nonzero.

We recall the local intersection bound for this restriction. In a
one-dimensional Noetherian local domain $(R,\frakm_R)$, let
$0\ne f\in\frakm_R^m$. For every integer $q\ge1$, multiplication by
powers of $f$ gives
\[
 \length(R/(f^q))=q\,\length(R/(f)).
\]
Since $(f^q)\subset\frakm_R^{mq}$, the Hilbert--Samuel formula yields
\[
 q\,\length(R/(f))
 \ge\length(R/\frakm_R^{mq})
 =mq\,e(R)+O(1),
\]
where $e(R)$ is the ordinary multiplicity. Dividing by $q$ and letting
$q\to\infty$ gives
\begin{equation}
 \label{eq:local-curve-bound}
 \length(R/(f))\ge m\,e(R).
\end{equation}

Apply \eqref{eq:local-curve-bound} to the local equations of $s|_C$ at
the marked points on $C$. The zero scheme of this nonzero section is an
effective Cartier divisor on the integral projective curve, and its total
length is $\deg(A|_C)=A\cdot C$. Summing the local contributions proves
the assertion.
\end{proof}

Fix a tuple $\mathbf p\in U_r(X)\setminus Z_r$. Applying
Lemma~\ref{lem:jets-curves} with $A=kL$ and
$m=\lfloor k\theta_j\rfloor$ gives
\[
 \eps(X,L;\mathbf p)\ge\frac{\lfloor k\theta_j\rfloor}{k}
 \qquad(k\ge k_j).
\]
First let $k\to\infty$ for each fixed $j$, and then let $j\to\infty$.
We obtain
\begin{equation}
 \label{eq:main-lower-bound}
 \eps(X,L;\mathbf p)\ge w=(V/r)^{1/n}.
\end{equation}

\subsection{The volume bound}

For completeness, we derive the reverse inequality for every tuple of
distinct points. Let $\pi:Y\to X$ be the blow-up of such a tuple, with
exceptional divisors $E_1,\ldots,E_r$. For $t\ge0$, put
\[
 D_t=\pi^*L-t\sum_{i=1}^r E_i.
\]
If an integral curve $\Gamma$ is contained in $E_i\simeq\mathbb P^{n-1}$,
then $D_t|_{E_i}=t\mathcal O_{\mathbb P^{n-1}}(1)$ is nonnegative on
$\Gamma$. Every other integral curve is the strict transform of an
integral curve $C\subset X$, and
\begin{equation}
 \label{eq:strict-transform-intersection}
 D_t\cdot\Gamma=L\cdot C-t\sum_{i=1}^r\mult_{p_i}C.
\end{equation}
Here $E_i\cdot\Gamma=\mult_{p_i}C$ also when $C$ is singular. If
$p_i\notin C$, both sides vanish. For $p_i\in C$, identify the strict
transform locally with the blow-up of $\mathcal O_{C,p_i}$ at its maximal
ideal. Its intersection
with $E_i$ is the zero-dimensional exceptional fiber
$\operatorname{Proj}(\operatorname{gr}_{\frakm}\mathcal O_{C,p_i})$,
whose length is the Hilbert--Samuel multiplicity. This is the degree of
the effective Cartier divisor cut out by $E_i$ on the strict transform.

It follows from \eqref{eq:strict-transform-intersection} that $D_t$ is
nef whenever $0\le t<\eps(X,L;\mathbf p)$. A nef real divisor has
nonnegative top self-intersection. The exceptional divisors are disjoint,
$\pi^*L|_{E_i}$ is trivial, and $E_i^n=(-1)^{n-1}$. Therefore
\[
 0\le D_t^n=V-rt^n.
\]
This inequality for all $0\le t<\eps(X,L;\mathbf p)$ proves the universal
bound
\begin{equation}
 \label{eq:volume-upper-bound}
 \eps(X,L;\mathbf p)\le(V/r)^{1/n}.
\end{equation}
Together, \eqref{eq:main-lower-bound} and \eqref{eq:volume-upper-bound}
give the equality asserted in Theorem~\ref{thm:main} at every tuple outside
$Z_r$. Equation~\eqref{eq:strict-transform-intersection} also shows that
$D_w$ is nef there, and its top self-intersection is zero. The threshold
\eqref{eq:threshold} was chosen before fixing $r$, so this proves the
theorem for every integer $r\ge r_0$.
==== END Maximal-Multipoint-Seshadri-Constants-in-Higher-Dimensions//conclusion.tex ====

==== BEGIN Maximal-Multipoint-Seshadri-Constants-in-Higher-Dimensions//flag.tex ====
\section{A simplex from a complete-intersection flag}
\label{sec:flag}

We now associate to $(X,L)$ a graded support family with the same dimensions
as the spaces of sections. Its exponent sets lie in a simplex whose volume
is exactly the leading coefficient of the Hilbert polynomial. This equality
of volumes will later ensure that changing the shape of the simplex loses
no asymptotic supply of exponents.

Encoding sections by the least exponents associated with a flag is standard
in the theory of Okounkov bodies; see, for example,
\citet[Lemma~1.3]{LazarsfeldMustata2009}. For our complete-intersection flag,
the required bound follows directly from division by its defining sections
and a degree calculation on the final curve.

For positive real numbers $a_1,\ldots,a_n$, write
\begin{equation}\label{eq:simplex-definition}
 \Delta(a_1,\ldots,a_n)
 =\left\{y\in\R_{\geq0}^n:
              \sum_{i=1}^n\frac{y_i}{a_i}\leq1\right\}.
\end{equation}
Its Euclidean volume is $a_1\cdots a_n/n!$.

\begin{proposition}[The initial simplex]\label{prop:flag-simplex}
Let $X$ be a smooth integral complex projective variety of dimension
$n\geq3$, let $L$ be ample, and put $V=L^n$. Choose a positive integer $D$
such that $DL$ is very ample. There is a graded support family
$S_k\subset\Z_{\geq0}^n$ such that
\begin{align}
 S_k&\subset kP_0,
 &P_0&=\Delta(1/D,\ldots,1/D,D^{n-1}V),
 \label{eq:flag-simplex}\\
 \#S_k&=h^0(X,kL)
       =\frac{V}{n!}k^n+O(k^{n-1}),
 &\vol(P_0)&=\frac{V}{n!}.
 \label{eq:flag-count}
\end{align}
For all integers $k\geq1$, $m\geq0$, and $r\geq1$, if
$\M(S_k)$ separates jets through degree $m$ at some distinct $r$-tuple
in $\Torus^n$, then $H^0(X,kL)$ does so at some distinct $r$-tuple in $X$.
\end{proposition}

\begin{proof}
Choose general sections $\sigma_1,\ldots,\sigma_{n-1}$ of $DL$ so that
their successive zero loci form a smooth integral flag
\[
 X=X_0\supset X_1\supset\cdots\supset X_{n-1}=C,
 \qquad
 X_i=X_{i-1}\cap\{\sigma_i=0\}.
\]
Such a flag exists by Bertini's theorem for very ample hyperplane sections.
Choose $p\in C$ and a local frame $e$ of $L$ near $p$. The functions
$z_i=\sigma_i/e^D$ for $1\leq i<n$ have independent differentials at $p$;
complete them by a function $z_n$ to local analytic coordinates centered
at $p$.

Represent every section of $kL$ by its germ in the frame $e^k$. These
representations are compatible with products in different degrees. They
are injective, since a section on an integral variety vanishing on an open
germ vanishes identically. Order Taylor exponents lexicographically,
starting with the exponent of $z_1$, and let $S_k$ be the set of possible
least exponents of nonzero section germs. Lemma~\ref{lem:least-terms}
gives
\[
 \#S_k=h^0(X,kL),\qquad S_k+S_l\subset S_{k+l}.
\]

We next bound these exponents by global vanishing orders. Let
$\alpha=(\alpha_1,\ldots,\alpha_n)\in S_k$ be the least exponent of a
section $s$. Put $s_0=s$. For $1\leq i<n$, suppose inductively that the residual section on
$X_{i-1}$ has least exponent $(\alpha_i,\ldots,\alpha_n)$. Its local
order along $X_i$ is $\alpha_i$. A restriction to the integral divisor
$X_i$ that vanishes near $p$ vanishes identically. The Cartier divisor
exact sequence therefore allows division by
$\sigma_i|_{X_{i-1}}$, one factor at a time, exactly $\alpha_i$ times.
Define
\[
 s_i=
 \left.
 \frac{s_{i-1}}{(\sigma_i|_{X_{i-1}})^{\alpha_i}}
 \right|_{X_i}
 \in H^0\left(X_i,
       \left(k-D\sum_{j=1}^i\alpha_j\right)L|_{X_i}\right).
\]
The exact local order ensures that $s_i$ is nonzero. Its germ is obtained
by dividing by $z_i^{\alpha_i}$ and setting $z_i=0$, so its least exponent
is $(\alpha_{i+1},\ldots,\alpha_n)$. This completes the induction. At
$i=n-1$ we obtain a nonzero section on $C$ of
\[
 \left(k-D\sum_{i=1}^{n-1}\alpha_i\right)L|_C.
\]
Its vanishing order at $p$ is $\alpha_n$. Since
$\deg(L|_C)=D^{n-1}V$, the degree bounds that order and yields
\[
 \alpha_n\leq
 \left(k-D\sum_{i=1}^{n-1}\alpha_i\right)D^{n-1}V.
\]
Equivalently,
\[
 D\sum_{i=1}^{n-1}\alpha_i+
 \frac{\alpha_n}{D^{n-1}V}\leq k,
\]
which proves \eqref{eq:flag-simplex}. The volume formula and the Hilbert
polynomial asymptotic for an ample line bundle give
\eqref{eq:flag-count}.

It remains to transfer jets from the least monomials to the sections. Fix
$k$. If $S_k$ is empty, the asserted implication has a false premise and
there is nothing to prove. Otherwise, choose a basis with distinct least
exponents. A single positive
weight vector exposes all these least terms, even though the series may
have infinite support. Indeed, put $\lambda_n=1$ and choose the preceding
weights recursively so that
\begin{equation}\label{eq:lex-exposing-weights}
 \lambda_i>
 \max_{\alpha\in S_k}\sum_{j>i}\lambda_j\alpha_j
 \qquad(1\leq i<n).
\end{equation}
If $\beta$ is lexicographically later than $\alpha$ and $i$ is their
first differing coordinate, then
\[
 \lambda\cdot(\beta-\alpha)
 \geq\lambda_i-\sum_{j>i}\lambda_j\alpha_j>0.
\]
Here we used that all Taylor exponents are nonnegative.

Now take a torus tuple at which $\M(S_k)$ separates the specified jets.
The substitution $z_i=s^{\lambda_i}Z_i$ sends this tuple into the chosen
coordinate chart for sufficiently small $s>0$, without identifying any
points. Lemma~\ref{lem:weighted-limit} gives the least monomials as limits
of normalized section germs, and Lemma~\ref{lem:rank-transfer} gives jet
separation by the original sections at the image tuple in $X$.
\end{proof}

The weights in the last paragraph may depend on $k$. In contrast, the flag,
the local frame, and the lexicographic order are fixed once and for all.
Thus Proposition~\ref{prop:flag-simplex} supplies a single graded family,
together with the separate finite-degree interpolation implications needed
to use it.
==== END Maximal-Multipoint-Seshadri-Constants-in-Higher-Dimensions//flag.tex ====

==== BEGIN Maximal-Multipoint-Seshadri-Constants-in-Higher-Dimensions//introduction.tex ====
\section{Introduction}\label{sec:introduction}

Let $S$ be a smooth integral complex projective surface and $L$ an ample
line bundle. We use additive notation for tensor powers and put $H=L^2$.
For a tuple of distinct points $\mathbf p=(p_1,\ldots,p_r)$, the multipoint
Seshadri constant is
\[
 \varepsilon(S,L;\mathbf p)
 =\inf_C\frac{L\cdot C}{\sum_{i=1}^r\mult_{p_i}C},
\]
where $C$ ranges over integral curves meeting at least one of the points.
If $\pi:Y\to S$ is the blow-up of these points, with exceptional curves
$E_1,\ldots,E_r$, the same constant is the largest $\lambda\ge0$ for
which $\pi^*L-\lambda\sum_iE_i$ is nef. Here a real divisor class is
\emph{nef} if it has nonnegative intersection with every integral curve.
This constant measures how much of the positivity of $L$ remains after
subtracting equal exceptional weights at the chosen points. Its universal
bound is
\begin{equation}\label{eq:upper-bound}
 \varepsilon(S,L;\mathbf p)\le\sqrt{\frac Hr};
\end{equation}
an elementary jet-count proof appears in Lemma~\ref{lem:sesha-upper}.

Write
\[
 U_r=\{(p_1,\ldots,p_r)\in S^r:p_i\ne p_j\text{ for }i\ne j\}.
\]
A property holds at \emph{very general} tuples if its failures are contained
in a countable union of proper Zariski-closed subsets of $U_r$.

\begin{theorem}\label{thm:main}
For every smooth integral complex projective surface $S$ and every ample
line bundle $L$ on $S$, there is an integer $r_0=r_0(S,L)$ with the following
property. For each integer $r\ge r_0$, there is a countable union
$Z_r$ of proper Zariski-closed subsets of $U_r$, with nonempty complement,
such that for every $\mathbf p\in U_r\setminus Z_r$ the class
\[
 \pi^*L-\sqrt{\frac{L^2}{r}}\sum_{i=1}^r E_i
\]
is nef on the blow-up at $\mathbf p$. Consequently,
\[
 \varepsilon(S,L;\mathbf p)=\sqrt{\frac{L^2}{r}}.
\]
\end{theorem}

Theorem~\ref{thm:main} resolves positively the qualitative Nagata--Biran
conjecture for surfaces, also called the qualitative
Nagata--Biran--Szemberg conjecture. The threshold is allowed to depend on
the polarized surface. The conclusion concerns the nef boundary class of
self-intersection zero. Thus it excludes every curve whose ratio of degree
to total point multiplicity falls below the numerical bound, even when
those multiplicities are unequal. The sharp threshold $r\ge10$ and strict nonhomogeneous
inequality in the plane Nagata problem at very general complex points
are a separate refinement, proved in
\cite[Theorem~1.1]{OpenAINagata2026}.
Corollary~\ref{cor:fields} extends eventual maximality to uncountable
algebraically closed fields of characteristic zero.

\subsection{Background}

Nagata formulated his plane conjecture in his work on Hilbert's
fourteenth problem and proved its strict multiplicity inequality when the
number of points is a square at least $16$
\cite{Nagata1959}; see also \cite[Chapter~III, Section~1, Proposition~1]{Nagata1965}. Multipoint Seshadri constants place that
problem in the geometry of arbitrary polarized surfaces. The qualitative
formulation asks for equality in \eqref{eq:upper-bound} for every
sufficiently large integer $r$; see
Syzdek--Szemberg~\cite[Conjecture~4.3]{SyzdekSzemberg2010}.
Strycharz-Szemberg--Szemberg~\cite[Conjecture~2.1]{StrycharzSzemberg2004}
formulate a stronger prediction specifying the threshold in terms of a
linear system on the surface.

Biran's stability theorem gives full symplectic packings for sufficiently
large numbers of equal balls, for rational symplectic classes on closed
four-manifolds \cite{Biran1999}; see also
\cite[Theorem~6.3]{Biran2001}. Buse--Hind--Opshtein subsequently proved
strong packing stability, allowing unequal sufficiently small balls, for
all closed symplectic four-manifolds
\cite[Theorem~1]{BuseHindOpshtein2016}. The algebraic statement asks for
nefness on blow-ups of the fixed complex surface. Its packing interpretation
uses K\"ahler forms, as made precise by
Eckl \cite[Theorem~0.5]{Eckl2017}; symplectic packing stability alone
does not fix the required complex structure.

On the algebraic side, Harbourne's estimates include maximality for all
sufficiently large $r$ with $rL^2$ a square, as well as asymptotic lower
bounds \cite[Theorem~I.1 and Proposition~I.2]{Harbourne2003}. The stated
results assume very ampleness; for an ample $L$, apply them to a very ample
power $aL$ and use $\varepsilon(S,aL;\mathbf p)=a\varepsilon(S,L;\mathbf p)$.
For these square values of $rL^2$, intersecting over $\C$ the countably
many open loci for bounds approaching the maximum gives equality at very
general tuples.
Assuming maximality for the corresponding plane constant, Ro\'e's transfer
gives the maximal bound when the surface
has a point with maximal one-point Seshadri constant
\cite[Theorem~3 and Corollary~7]{Roe2004}.
The multipoint product inequality, due to Biran for surfaces and stated
in greater generality by Ro\'e--Ross, propagates maximality from a fixed
configuration to suitable multiples of its size when the corresponding
plane constants are maximal
\cite[Theorem~1.1 and Remark~1.2]{RoeRoss2009}.
Our argument treats arbitrary ample polarizations and all integers beyond
one threshold directly.

There is also a distinction between eventual maximality and exact values
on particular surfaces. Dionne--Roth compute multipoint constants for
substantial families of polarizations on $\PP^1\times\PP^1$
\cite{DionneRoth2025}. The recent preprint of Laface--Ugaglia
\cite[Theorem~1]{LafaceUgaglia2026} claims the maximal irrational value
$1/\sqrt5$ for $\mathcal O(1,1)$ at ten very general points of that surface.
These fixed-surface and fixed-point-count results address a different
question from the uniform eventuality in Theorem~\ref{thm:main}.

\subsection{Proof overview}

The proof turns the problem into finite interpolation on the algebraic
torus $\T$. An \emph{injective jet test of order $m$} on a vector space
of functions means that only the zero function vanishes to order at least
$m$ at all the test points. We will show that, for every sufficiently
large $r$ and all positive integers $k,m$ with $m>k\sqrt{H/r}$, the
sections of $kL$ admit such a test at some $r$ distinct surface points.
The threshold in $r$ must work for arbitrarily large jet orders. This
uniformity is different from the asymptotic postulation theorem of
Alexander--Hirschowitz \cite[Theorem~1.1 and Corollary~1.2]{AlexanderHirschowitz2000},
where the degree threshold depends on a fixed bound for the point
multiplicities.

Section~\ref{sec:jets} establishes the transfer principle: a full-rank
jet matrix for the initial Taylor monomials remains full rank after a
sufficiently small analytic rescaling. We then choose one integral divisor
$V\in|dL|$ with an ordinary node.
In local coordinates its two branches are the coordinate axes. The
intersection number $kL\cdot V$ bounds the least Taylor exponent of a
section under weights $(1,t)$, after any copies of $V$ are removed.
The resulting exponents lie in a quadrilateral $kQ_t$.
A change of torus coordinates, followed by compression of the slices
parallel to $(1,1)$, gives a triangle $kP_t$ of area
$k^2H/2$ (Section~\ref{sec:node}). The slice compression has its own finite
jet-matrix limit. Applying these two transfers in succession carries an
injective torus test for this triangle to sections on the surface.
Monomial diagrams and nonzero interpolation minors are central in
Dumnicki's diagram method \cite[Proposition~13 and Theorem~14]{Dumnicki2007};
the particular weighted-exponent bound and diagonal compression needed
here are proved in full.

The next step uses the continuously varying shape of $P_t$. The equality
$\operatorname{area}(P_t)=H/2$ makes the torus jet threshold exactly
$\sqrt{H/r}$.
For every sufficiently large integer $r$, we tune this shape to a primitive
lattice direction and choose a subgroup of $\T$ with exactly $r$ points.
The subgroup is the kernel of two monomial equations with exponent
lattice of index $r$; it need not be a square grid. The parameter $t$ and
the subgroup are fixed before the jet orders $k,m$ are chosen.
Projection onto its characters reduces the jet test to a Laurent polynomial
supported in a triangle of area $(kw)^2/2$ and vertical range $kw$, where
$w=\sqrt{H/r}$. An elementary estimate using horizontal slices bounds its
multiplicity at the identity by $kw$ (Section~\ref{sec:lattice}).

Finally, the transferred surface tuple can depend on $k,m$, but injectivity
is a Zariski-open rank condition. Countably many such conditions give
simultaneous tests at very general tuples. To pass from equal vanishing
orders to all curves, suppose that a curve has negative intersection with
the square-zero boundary class. Subtracting a small multiple of that curve
and making a rational perturbation produces a positive-square divisor
beyond the boundary. Riemann--Roch makes a multiple effective; adding back
the curve yields a section with equal vanishing orders larger than the
critical slope, contradicting the jet tests (Section~\ref{sec:nef}).
==== END Maximal-Multipoint-Seshadri-Constants-in-Higher-Dimensions//introduction.tex ====

==== BEGIN Maximal-Multipoint-Seshadri-Constants-in-Higher-Dimensions//jets.tex ====
\section{Finite jet tests and initial monomials}\label{sec:jets}

We first explain how an injective jet test for initial monomials gives an
injective jet test for the original holomorphic functions.  The argument
uses finite matrices and convergent power series.

For a holomorphic function $f$ near a point $q$ of a smooth complex surface,
write $\ord_q(f)$ for its order of vanishing, with $\ord_q(0)=\infty$.
In local coordinates, $\ord_q(f)\ge m$ means that all partial derivatives of
total order less than $m$ vanish at $q$.  For distinct points
$\boldsymbol q=(q_1,\ldots,q_r)$ and a finite-dimensional space $F$ of
functions defined near them, its \emph{order-$m$ jet map} is
\[
 J^m_{\boldsymbol q}\colon F\longrightarrow
 \bigoplus_{j=1}^r\mathcal O_{q_j}/\mathfrak m_{q_j}^{m},
 \qquad f\longmapsto (f\bmod\mathfrak m_{q_j}^{m})_{j=1}^r.
\]
Here $\mathcal O_{q_j}$ is the ring of holomorphic germs at $q_j$, and
$\mathfrak m_{q_j}$ is its ideal of germs vanishing there.  We call the
corresponding jet test \emph{injective} when this map is injective, or
equivalently when no nonzero $f\in F$ vanishes to order at least $m$ at
every $q_j$.  In two local coordinates, the matrix of this map records
the derivatives of total order less than $m$ of a basis of $F$; it has
$r\binom{m+1}{2}$ rows.  Its injectivity is exactly full column rank.

The same definition applies to sections of a line bundle using local
frames.  Multiplication by a holomorphic unit induces an invertible map
on each jet quotient, so the definition is independent of those frames.
A biholomorphic change of coordinates preserves the powers of the
maximal ideal and hence preserves orders of vanishing and jet
injectivity.

Finite matrices also underlie monomial diagram methods for plane
interpolation \cite[Propositions~10 and~13]{Dumnicki2007}. The transfer
below applies to convergent local functions and follows from persistence
of one nonzero maximal minor.

\begin{lemma}[Persistence of a finite jet test]\label{lem:rank-persistence}
Let $m,r,N$ be positive integers and let $q_1,\ldots,q_r$ be distinct
points of an open subset $U\subset\C^2$.  For $1\le i\le N$ and
$0\le s<s_0$, let $f_{i,s}$ be holomorphic on $U$.  Suppose that, as
$s\downarrow0$, every derivative of $f_{i,s}$ of total order less than
$m$ at each $q_j$ converges to the corresponding derivative of $f_{i,0}$.
If the jet matrix of $f_{1,0},\ldots,f_{N,0}$ has column rank $N$, then
the jet matrix of $f_{1,s},\ldots,f_{N,s}$ also has column rank $N$ for
all sufficiently small positive $s$.

In particular, convergence locally uniformly on $U$ suffices for the
derivative hypothesis.
\end{lemma}

\begin{proof}
A nonzero $N\times N$ minor of the limiting jet matrix remains nonzero
for all sufficiently small $s$, since each of its entries converges.
The final assertion follows from the Cauchy integral formula on small
polydiscs around the finitely many points.
\end{proof}

For a bounded set $\Omega\subset\R^2$, denote by
\[
 \mathcal M(\Omega)
 =\Span_{\C}\{x^\alpha z^\beta:
                 (\alpha,\beta)\in\Omega\cap\Z^2\}
\]
the finite-dimensional space of Laurent polynomials on the torus
$\T$.  When another pair of torus coordinates is specified,
the same notation uses that pair in place of $(x,z)$.

We now consider convergent power series at the origin.  Fix $t>0$ and
order pairs $(\alpha,\beta)\in\Z_{\ge0}^2$ lexicographically by
\[
 (\alpha+t\beta,\beta).
\]
Every bounded interval of weights contains only finitely many exponent
pairs.  Thus each nonzero series $f\in\C\{x,z\}$ has a first nonzero
exponent in this order, denoted by $\nu_t(f)$.  For a finite-dimensional
subspace $F\subset\C\{x,z\}$, put
\[
 E_t(F)=\{\nu_t(f):0\ne f\in F\}.
\]

\begin{proposition}[Transfer from initial monomials]
\label{prop:initial-transfer}
Let $F\subset\C\{x,z\}$ be a finite-dimensional complex vector space
of convergent holomorphic germs, and let $t>0$.  The set $E_t(F)$ has
exactly $\dim F$ elements, and $F$ admits a basis whose initial exponents
are these distinct elements.

Let $r,m$ be positive integers.  If the monomial space
\[
 \Span_{\C}\{X^\alpha Z^\beta:(\alpha,\beta)\in E_t(F)\}
\]
has an injective order-$m$ jet test at some distinct points
$q_1,\ldots,q_r\in\T$, then representatives of $F$ on a common
neighborhood of the origin have an injective order-$m$ jet test at
distinct points arbitrarily close to the origin.
\end{proposition}

\begin{proof}
The statement is immediate if $F=0$, so write $N=\dim F>0$.

\emph{Choose a basis with distinct initial terms.}
Among the initial exponents of nonzero elements of $F$, choose the
least one, $e_1$.  Every element of $F$ has zero coefficient at every
earlier exponent.  The coefficient at $e_1$ is a nonzero linear
functional on $F$.  Choose $f_1$ on which it equals $1$, and continue
inside its codimension-one kernel.  After $N$ steps this gives a basis
$f_1,\ldots,f_N$ with strictly increasing initial exponents
$e_i=(\alpha_i,\beta_i)$ and initial coefficients $1$.  In a nonzero
linear combination of the basis vectors, the first vector with nonzero
coefficient supplies its uncancelled initial term.  Consequently
$E_t(F)=\{e_1,\ldots,e_N\}$.

\emph{Separate the initial terms by a single positive weight.}
Set
\[
 M=1+\max_{1\le i\le N}(\alpha_i+t\beta_i).
\]
Choose a small number $\eta>0$.  For $\eta$ sufficiently small, all
the weights $\alpha_i+(t+\eta)\beta_i$ remain below $M$.  Every term
of old weight at least $M$ has new weight at least $M$, since its second
exponent is nonnegative, so it cannot become a least term.

Only finitely many exponent pairs have old weight below $M$.  For a
noninitial term of $f_i$ in this finite set, either its old weight is
strictly greater than that of $e_i$, or its old weight is equal and its
second exponent is strictly greater.  In the first case the strict
inequality persists for all sufficiently small $\eta>0$.  In the second
case increasing $t$ by $\eta$ makes the new weight strictly greater.
One common choice of $\eta>0$ therefore makes $e_i$ the unique term of
least weight $(1,t')$ in every $f_i$, where $t'=t+\eta$.  Put
\[
 \lambda_i=\alpha_i+t'\beta_i.
\]

\emph{Dilate the convergent series.}
Choose representatives of the finitely many basis germs on a common
polydisc, and define, for positive real $s$,
\[
 g_{i,s}(X,Z)=s^{-\lambda_i}f_i(sX,s^{t'}Z).
\]
These functions converge locally uniformly on $\C^2$ to
$X^{\alpha_i}Z^{\beta_i}$, in the sense that they are defined on any
fixed compact set for all sufficiently small $s$.  To verify the
convergence, write
$f_i(x,z)=\sum_{\alpha,\beta}c_{i,\alpha,\beta}x^\alpha z^\beta$.
Fix $R>0$ and choose $0<s_0<1$ so that the radii
$(s_0R,s_0^{t'}R)$ lie strictly inside a common polydisc of absolute
Taylor convergence.  Every supported exponent satisfies
$\alpha+t'\beta-\lambda_i\ge0$, with equality only at $e_i$.
For $0<s\le s_0$, the sum of uniform majorants for the rescaled terms
on $|X|,|Z|\le R$ is
\[
 \sum_{\alpha,\beta}
 |c_{i,\alpha,\beta}|s_0^{\alpha+t'\beta-\lambda_i}R^{\alpha+\beta}
 =s_0^{-\lambda_i}
   \sum_{\alpha,\beta}|c_{i,\alpha,\beta}|
     (s_0R)^\alpha(s_0^{t'}R)^\beta
 <\infty.
\]
Each noninitial term tends uniformly to zero.  The summable majorant
therefore proves the asserted uniform convergence.  Applying this
argument on a slightly larger polydisc and using the Cauchy integral
formula gives convergence of all derivatives of any fixed finite
order on the smaller polydisc.

\emph{Transfer the jet test.}
At the given points $q_j$, the limiting monomials have an injective jet
test.  Lemma~\ref{lem:rank-persistence} shows that the functions
$g_{1,s},\ldots,g_{N,s}$ have an injective jet test at the same points
for every sufficiently small positive $s$.  The map
\[
 D_s(X,Z)=(sX,s^{t'}Z)
\]
is invertible for each such $s$.  Its images $p_j=D_s(q_j)$ are distinct,
lie in the common coordinate neighborhood, and approach the origin.
Pullback by $D_s$ preserves orders of vanishing at corresponding
points.  The nonzero scalar factors $s^{-\lambda_i}$ change only the
basis.  Hence the original space $F$ has an injective order-$m$ jet
test at $p_1,\ldots,p_r$.
\end{proof}

The auxiliary weight $t'$ in this proof separates finitely many initial
terms chosen using $t$; those exponents remain unchanged.  Thus a bound
for $E_t(F)$ in a fixed diagram is preserved throughout the argument.
The weights and the positive real dilation parameter need not be
rational, and the construction makes no assertion about their uniformity
over different finite-dimensional spaces.

To apply Proposition~\ref{prop:initial-transfer} to global sections on
$S$, choose local coordinates and a nonvanishing frame for the line
bundle.  Restriction to germs is injective: a section vanishing on a
neighborhood vanishes everywhere by the identity principle on the
connected surface.  A change of frame multiplies each germ by a
holomorphic unit, whose nonzero constant term preserves its initial
exponent.  In particular, if all the initial exponents lie in a bounded
set $\Omega\subset\R_{\ge0}^2$, an injective jet test for the larger
space $\mathcal M(\Omega)$ suffices for the transfer.
==== END Maximal-Multipoint-Seshadri-Constants-in-Higher-Dimensions//jets.tex ====

==== BEGIN Maximal-Multipoint-Seshadri-Constants-in-Higher-Dimensions//lattice.tex ====
\section{Finite subgroups as jet tests}\label{sec:lattice}

We now find an injective jet test for the triangle spaces
$\mathcal M(kP_t)$.  The test points will form a finite subgroup of the
complex torus.  Averaging over this group separates a Laurent polynomial
into components supported on cosets of a sublattice.  In the coordinates
of that sublattice, each component lies in a triangle for which a
one-point multiplicity estimate applies.

\subsection{A multiplicity estimate for triangles}

For a bounded subset of $\R^2$, its \emph{vertical range} is the length of
its projection onto the second coordinate axis.  The following estimate
allows arbitrary real vertices and translations.

\begin{lemma}\label{lem:triangle-multiplicity}
Let $\rho>0$, and let $\Delta\subset\R^2$ be a triangle of area
$\rho^2/2$ and vertical range $\rho$.  Every nonzero Laurent polynomial
$F\in\C[U^{\pm1},Z^{\pm1}]$ whose exponent support is contained in
$\Delta$ satisfies
\[
 \ord_{(1,1)}F\leq\rho.
\]
\end{lemma}

\input{figures/triangle}

\begin{proof}
For a height $y$ in the vertical projection of $\Delta$, let $\ell(y)$
be the horizontal length of the slice of $\Delta$ at that height.  Write
$y_0\leq y_1\leq y_2$ for the three vertex heights; thus
$y_2-y_0=\rho$.  If these heights are distinct, $\ell$ increases linearly
from zero at $y_0$ to its maximum at $y_1$, then decreases linearly to
zero at $y_2$.  If two vertex heights coincide, $\Delta$ has a horizontal
side and there is just one linear segment.  In either case, the area is
one half the vertical range times the maximum slice length.  The maximum
is therefore $\rho$.

More precisely, for $0\leq e\leq\rho$, the heights with
$\ell(y)\geq e$ form an interval of length $\rho-e$
(Figure~\ref{fig:triangle-slices}).  When the three
heights are distinct, its endpoints are
\[
 y_0+\frac e\rho(y_1-y_0),
 \qquad
 y_2-\frac e\rho(y_2-y_1).
\]
The same formulas apply when two adjacent heights coincide, including
the cases $e=0$ and $e=\rho$.

Write $F=\sum_q f_q(U)Z^q$, with each displayed row $f_q$ nonzero, and
let $e$ be the least order of these rows at $U=1$.  Dividing every row by
$(U-1)^e$ gives
\[
 F=(U-1)^eD,
 \qquad D=\sum_q d_q(U)Z^q,
 \qquad D(1,Z)\ne0.
\]
Indeed, at least one $d_q(1)$ is nonzero, and distinct powers of $Z$ are
linearly independent.  All $d_q$ remain Laurent polynomials, including
when some exponents are negative.

The horizontal exponent span of the row
$f_q=(U-1)^e d_q$ is exactly $e$ plus that of $d_q$: the extreme
coefficients of the product are nonzero.  Thus $\ell(q)\geq e$ for every
height occurring in $D$.  There is no cancellation between distinct
heights, since the multiplying factor depends only on $U$.  It follows
that $e\leq\rho$ and that the heights occurring in $D$ span an interval
of length at most $\rho-e$.

On setting $U=1$, some rows may disappear, but no new height appears.
The nonzero Laurent polynomial $D(1,Z)$ therefore has exponent span at
most $\rho-e$.  Multiplication by a power of $Z$, which is a unit at
$Z=1$, turns it into an ordinary polynomial of degree at most
$\rho-e$.  Consequently
\[
 \ord_{(1,1)}D
 \leq \ord_1 D(1,Z)
 \leq \rho-e.
\]
The first inequality holds because restriction cannot introduce a term
of smaller total degree into the local Taylor series.  Orders add under
products of nonzero holomorphic germs, so $\ord_{(1,1)}F\leq\rho$.
All exponent spans and orders used here are integers; no integrality of
$\rho$, the vertices, or the translating vector is required.
\end{proof}


\subsection{The subgroup test}

For integer vectors $u,\xi\in\Z^2$, write $\det(u,\xi)$ for their
oriented determinant, extending this notation linearly to real $\xi$.
A vector in $\Z^2$ is \emph{primitive} if its coordinates are coprime.

\begin{proposition}\label{prop:subgroup-test}
Let $H>0$, let $r$ be a positive integer, and let $P\subset\R^2$ be a
triangle of area $H/2$.  Suppose there is a primitive vector $u\in\Z^2$
such that
\[
 \max_{\xi\in P}\det(u,\xi)
 -\min_{\xi\in P}\det(u,\xi)=\sqrt{rH}.
\]
Then there is a subgroup $G\subset\T$ of exactly $r$ distinct points
such that, for all positive integers $k,m$ with
$m>k\sqrt{H/r}$, the order-$m$ jet test on $\mathcal M(kP)$ at $G$ is
injective.
\end{proposition}

\begin{proof}
Complete $u$ to an integer basis $(u,v_0)$ with $\det(u,v_0)=1$.
Put $v=rv_0$ and
\[
 M=\Z u+\Z v\subset\Z^2.
\]
This sublattice has index $r$.  For $\eta=(\eta_1,\eta_2)\in\Z^2$,
write $\chi_\eta(x,z)=x^{\eta_1}z^{\eta_2}$ for the corresponding
character on $\T$, and define
\[
 G=\{p\in\T:\chi_\eta(p)=1\text{ for every }\eta\in M\}.
\]
The characters $X=\chi_u$ and $Y=\chi_{v_0}$ are torus coordinates:
their exponent matrix is unimodular, so its inverse also has integer
entries.  In these coordinates,
\[
 G=\{(X,Y):X=1,\ Y^r=1\}.
\]
Over $\C$, this consists of exactly $r$ distinct points.

Suppose, toward a contradiction, that a nonzero
$f\in\mathcal M(kP)$ has order at least $m$ at every point of $G$.
For $g\in G$, let $(\tau_gf)(p)=f(gp)$.  Translation by $g$ permutes
$G$ and is locally invertible, so every $\tau_gf$ has the same prescribed
vanishing.  For a coset $\eta+M$, its character projection is
\[
 f_\eta=\frac1r\sum_{g\in G}\chi_\eta(g)^{-1}\tau_gf.
\]
This is exactly the sum of the monomials of $f$ with exponents in
$\eta+M$.  Indeed, if $\xi-\eta=n u+\ell v_0$, then
$\chi_{\xi-\eta}$ restricts to $\zeta^\ell$ on $G$.  The sum over the
$r$th roots of unity gives
\[
 \frac1r\sum_{g\in G}\chi_{\xi-\eta}(g)
 =\begin{cases}
 1,&\xi-\eta\in M,\\
 0,&\xi-\eta\notin M.
 \end{cases}
\]
Thus the displayed projection keeps precisely the required coset.
Each $f_\eta$ still has order at least $m$ at every point of $G$, and its
support remains inside $kP$.  Since the components sum to $f$, choose
one that is nonzero.

Multiplication by the Laurent monomial $\chi_{-\eta}$ preserves local
orders and shifts this component's support into $M$.  There is therefore
a nonzero Laurent polynomial $F$ such that
\[
 \chi_{-\eta}f_\eta=F(U,Z),
 \qquad U=\chi_u,
 \qquad Z=\chi_v=Y^r.
\]
The map $(x,z)\mapsto(U,Z)$ is locally invertible at the identity:
in coordinates $(X,Y)$ it is $(X,Y)\mapsto(X,Y^r)$, whose derivative at
$(1,1)$ is $\operatorname{diag}(1,r)$.  The complex inverse function theorem therefore makes it a biholomorphism
on sufficiently small analytic neighborhoods.  In particular,
\[
 \ord_{(1,1)}F
 =\ord_{(1,1)}(\chi_{-\eta}f_\eta)\geq m.
\]
Only local invertibility is used; the map need not be injective on the
whole torus.

Let $N$ be the matrix with columns $u,v$.  The exponent support of $F$
lies in the real triangle
\[
 \Delta=N^{-1}(kP-\eta).
\]
Since $\det N=r$, this triangle has area $k^2H/(2r)$.  The second
coordinate of $N^{-1}\xi$ is $\det(u,\xi)/r$, so its vertical range is
$k\sqrt{rH}/r=k\sqrt{H/r}$.  Translation by $\eta$ changes neither
calculation.  Lemma~\ref{lem:triangle-multiplicity}, with
$\rho=k\sqrt{H/r}$, now gives
$\ord_{(1,1)}F\leq k\sqrt{H/r}<m$, the required contradiction.
\end{proof}

\subsection{A direction for every sufficiently large integer}

We apply Proposition~\ref{prop:subgroup-test} to
\[
 P_t=\conv\{(-b,0),(a,0),(0,1/d)\},
 \qquad
 a=\frac{dHt}{1+t},\quad b=\frac{dH}{1+t},\quad t>T,
\]
with $d,H,T$ fixed in Section~\ref{sec:node}.  Every triangle in this
family has area $H/2$, since $a+b=dH$.  Varying $t$ continuously will
make its determinant projection have the exact length needed for each
large $r$.

\begin{lemma}\label{lem:direction}
Fix $d,H>0$ and $T>1$, and define $P_t$ as above for $t>T$.
For every sufficiently large integer $r$ there are $t>T$ and a primitive
vector $u\in\Z^2$ such that the range of $\det(u,\cdot)$ on $P_t$ has
length $\sqrt{rH}$.  The threshold depends only on $d,H,T$.
\end{lemma}

\begin{proof}
Put $W=dH$ and $a_0=WT/(1+T)$.  As $t$ ranges over $(T,\infty)$,
$a=Wt/(1+t)$ ranges continuously and bijectively over $(a_0,W)$.
For $R=\sqrt{rH}$ sufficiently large, choose $j$ to be the largest power
of two at most $R/(2W)$.  Then
\[
 \frac{R}{4W}<j\leq\frac{R}{2W},
 \qquad jW<R.
\]
The open interval
\[
 I_r=\bigl(d(R-jW),\ d(R-ja_0)\bigr)
\]
has length
\[
 dj(W-a_0)>\frac{dR}{4(1+T)},
\]
which exceeds $2$ for every sufficiently large $r$.  Hence $I_r$
contains an odd integer $i$.  Its lower endpoint is positive, so $i>0$;
as $j$ is a power of two, the vector $u=(i,j)$ is primitive.

The strict inequalities defining $I_r$ imply
\[
 a_*:=\frac{R-i/d}{j}\in(a_0,W).
\]
Choose the unique $t>T$ for which $a=a_*$.  Since $b=W-a$, we have
\[
 \frac{i}{d}+ja=R,
 \qquad
 \frac{i}{d}-jb=R-jW>0.
\]
The determinant functional has the three vertex values
\[
 \det\bigl(u,(-b,0)\bigr)=jb,
 \qquad
 \det\bigl(u,(a,0)\bigr)=-ja,
 \qquad
 \det\bigl(u,(0,1/d)\bigr)=\frac{i}{d}.
\]
Its minimum is therefore $-ja$ and its maximum is $i/d$, giving the
required range $R$ exactly.  All lower bounds on $r$ in this construction
depend only on the fixed data $d,H,T$.
\end{proof}

\begin{corollary}\label{cor:torus-test}
For the fixed parameters $d,H,T$ of Section~\ref{sec:node}, there is an
integer $r_0$ such that for every integer $r\geq r_0$ one can choose a
single $t>T$ and a subgroup $G\subset\T$ of $r$ distinct points for
which the following holds: for all positive integers $k,m$ satisfying
\[
 m>k\sqrt{H/r},
\]
the order-$m$ jet test on $\mathcal M(kP_t)$ at $G$ is injective.
\end{corollary}

\begin{proof}
First fix $r$ and choose $t$ and $u$ by Lemma~\ref{lem:direction}.
The area of $P_t$ is $H/2$, so Proposition~\ref{prop:subgroup-test}
produces $G$ with the stated property simultaneously for all $k,m$.
Thus $r_0$, and then $t,G$, are chosen before these two jet parameters.
\end{proof}

Corollary~\ref{cor:surface-transfer} now gives a surface configuration
with the same injective jet test for each pair $k,m$.  These configurations
may depend on the pair.  The next section passes to very general points
where all the tests hold at once, and uses them to prove nefness.
==== END Maximal-Multipoint-Seshadri-Constants-in-Higher-Dimensions//lattice.tex ====

==== BEGIN Maximal-Multipoint-Seshadri-Constants-in-Higher-Dimensions//main.tex ====
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\hypersetup{pdftitle={Maximal Multipoint Seshadri Constants in Higher Dimensions},pdfauthor={OpenAI}}
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\title{Maximal Multipoint Seshadri Constants\protect\\ in Higher Dimensions}
\author{OpenAI}
\date{October 5, 2026}
\begin{document}
\maketitle
\begin{abstract}
Let $L$ be an ample line bundle on a smooth integral complex projective variety
$X$ of dimension $n\ge3$. We prove that there is a threshold
$r_0=r_0(X,L)$ such that for every integer $r\ge r_0$, the ordinary multipoint
Seshadri constant at $r$ very general points equals the volume bound
$(L^n/r)^{1/n}$. This establishes the qualitative Nagata--Biran--Szemberg
assertion in these dimensions.
\end{abstract}
\input{sections/introduction}
\input{sections/transfers}
\input{sections/flag}
\input{sections/reshaping}
\input{sections/saturation}
\input{sections/conclusion}
\bibliographystyle{plainnat}
\bibliography{references}
\end{document}
==== END Maximal-Multipoint-Seshadri-Constants-in-Higher-Dimensions//main.tex ====

==== BEGIN Maximal-Multipoint-Seshadri-Constants-in-Higher-Dimensions//nef.tex ====
\section{From homogeneous jets to nefness}\label{sec:nef}

The preceding sections produce an injective jet test at some tuple for
each pair of integers $k,m$ above the critical slope. We first make all
these tests hold at one very general tuple. A separate surface argument
then converts them into inequalities for every curve, including curves
whose multiplicities at the points are unequal.

\begin{lemma}[The open locus of an injective jet test]\label{lem:jet-open}
Let $S$ be a smooth integral complex projective surface, let $L$ be a line
bundle, and let $r,k,m$ be positive integers. In the configuration space
\[
 U_r=\{(p_1,\ldots,p_r)\in S^r:p_i\ne p_j\text{ for }i\ne j\},
\]
the tuples at which the order-$m$ jet test on $H^0(S,L^{\otimes k})$
is injective form a Zariski-open subset. If each test in a countable
collection with fixed $S,L,r$ is injective at some tuple, then all those tests are
simultaneously injective at very general tuples. Their common locus is
nonempty.
\end{lemma}

\begin{proof}
Let $q_1,q_2:S\times S\to S$ be the projections and let
$\mathcal I_\Delta$ be the ideal of the diagonal. The bundle of jets
of orders less than $m$ is
\[
 \mathcal J_{k,m}
 =q_{1*}\!\left(q_2^*L^{\otimes k}\otimes
       \mathcal O_{S\times S}/\mathcal I_\Delta^m\right).
\]
This is the algebraic principal-parts construction
\cite[Tags 0G3P and 0638]{Stacks}; it is locally free of rank
$\binom{m+1}{2}$.
Indeed, the diagonal is a regular embedding with conormal bundle
$\Omega_S^1$; the successive quotients of the displayed sheaf, pushed
forward to $S$, are
$L^{\otimes k}\otimes\operatorname{Sym}^j\Omega_S^1$ for
$0\le j<m$. They are locally free of ranks $j+1$.
The thickened diagonal is finite over $S$, so these pushforwards are
exact and give the asserted filtration. The fiber at $p$ is
$(L^{\otimes k})_p/\mathfrak m_p^m(L^{\otimes k})_p$.

Writing $\operatorname{pr}_i:U_r\to S$ for the coordinate projections,
consider the algebraic bundle map
\[
 H^0(S,L^{\otimes k})\otimes\mathcal O_{U_r}
 \longrightarrow\bigoplus_{i=1}^r\operatorname{pr}_i^*\mathcal J_{k,m}.
\]
Its fiber kernel consists exactly of sections vanishing to order at
least $m$ at every point. If $n=h^0(S,L^{\otimes k})>0$, failure of
injectivity is cut out locally by the $n\times n$ minors. If $n=0$,
the map is injective everywhere. In either case the injective locus
is Zariski open, and a single injective tuple makes it nonempty.
The analytic and algebraic finite jets agree at complex points:
in local coordinates both are the same Taylor polynomial modulo terms
of total degree at least $m$. Thus a tuple obtained by the analytic
transfers of the preceding sections is a valid witness here.

The variety $U_r$ is smooth and irreducible. A proper algebraic closed
subset has empty interior in its complex manifold of points. For a
countable collection of nonempty injective loci, their complements are
therefore closed nowhere-dense subsets of $U_r(\C)$. This locally
compact Hausdorff space is a Baire space, so their common complement
is nonempty, indeed dense. By construction it is the complement of a
countable union of proper Zariski-closed subsets of $U_r$, as required
for very general tuples.
\end{proof}

We next record the precise Riemann--Roch argument that will turn a
negative curve into an effective divisor with equal prescribed
multiplicities. The divisor being made effective need not be nef.

\begin{lemma}[Effective multiples of a positive-square divisor]
\label{lem:positive-square}
Let $Y$ be a smooth integral complex projective surface, let $A$ be a
nef real divisor class, and let $D$ be a rational Cartier divisor.
If $D^2>0$ and $A\cdot D>0$, then
$H^0(Y,\mathcal O_Y(kD))\ne0$ for every sufficiently large positive
integer $k$ divisible by the denominators of $D$.
\end{lemma}

\begin{proof}
Choose a positive integer $q$ such that $qD$ is Cartier and let
$k$ run through positive multiples of $q$. By Serre duality \cite[Tag 0FVV, Lemma 48.27.1(6)--(7)]{Stacks},
\[
 h^2(Y,\mathcal O_Y(kD))
   =h^0(Y,\mathcal O_Y(K_Y-kD)).
\]
For sufficiently large $k$,
\[
 A\cdot(K_Y-kD)=A\cdot K_Y-kA\cdot D<0.
\]
A divisor with a nonzero section is linearly equivalent to an
effective divisor, which has nonnegative intersection with the nef
class $A$. Hence the displayed $h^2$ vanishes for all such $k$.
Surface Riemann--Roch \cite[Tag 02UO]{Stacks} now gives
\[
 \begin{split}
 h^0(Y,\mathcal O_Y(kD))
 &\ge \chi(Y,\mathcal O_Y(kD))\\
 &=\frac{k^2D^2-kD\cdot K_Y}{2}+\chi(Y,\mathcal O_Y)>0
 \end{split}
\]
for sufficiently large $k$, since $D^2>0$.
\end{proof}

\begin{proposition}[Homogeneous tests imply nefness]
\label{prop:homogeneous-nef}
Let $S$ be a smooth integral complex projective surface, let $L$ be
ample, and let $p_1,\ldots,p_r$ be distinct points. Put
$H=L^2$ and $w=\sqrt{H/r}$. Suppose that, for all positive integers
$k,m$ with $m>kw$, no nonzero section of $L^{\otimes k}$ vanishes to
order at least $m$ at every $p_i$. On the blow-up
$\pi:Y\to S$ of these points, with exceptional curves $E_i$, the real
divisor class
\[
 D_0=\pi^*L-w\sum_{i=1}^rE_i
\]
is nef.
\end{proposition}

\begin{proof}
Choose a Cartier divisor representing $L$ and retain the notation $L$
for it. Write $A=\pi^*L$ and $E=\sum_iE_i$. The class $A$ is nef and
\[
 A^2=H,\qquad A\cdot E_i=0,\qquad
 E_i\cdot E_j=-\delta_{ij},\qquad D_0^2=0.
\]
If $D_0$ were not nef, there would be an integral curve $C\subset Y$
with $D_0\cdot C<0$. Since $D_0\cdot E_i=w>0$, this curve is not
exceptional, and $A\cdot C=L\cdot\pi_*C>0$.
For sufficiently small positive rational $\delta$, both
\[
 (D_0-\delta C)^2=-2\delta D_0\cdot C+\delta^2C^2>0,
 \qquad
 A\cdot(D_0-\delta C)=H-\delta A\cdot C>0
\]
hold. Fix such a $\delta$. By continuity, there is a rational number
$s>w$, sufficiently close to $w$, such that
\[
 D=A-sE-\delta C
 \quad\text{satisfies}\quad D^2>0,\qquad A\cdot D>0.
\]
Here $A\cdot D=H-\delta A\cdot C$ is independent of $s$.

Choose a sufficiently large positive integer $k$ divisible by the
denominators of both $s$ and $\delta$.
Lemma~\ref{lem:positive-square} supplies an effective divisor
$F\sim kD$. Since $k\delta$ and $m=ks$ are positive integers,
\[
 F+k\delta C\sim kA-mE
\]
is effective. The blow-up identity \cite[Tag 0AGP, Lemma 54.3.4(4)]{Stacks}
\[
 \pi_*\mathcal O_Y(-mE)
   =\bigcap_{i=1}^r\mathcal I_{p_i}^{\,m}
\]
and the projection formula give a nonzero section of
$L^{\otimes k}$ vanishing to order at least $m$ at every $p_i$.
To recall the local content of this identity, sections of a pullback
line bundle descend across the missing points because $S$ is normal,
and the exceptional order of a local equation on the blow-up of a
smooth point is its ordinary multiplicity. Thus the condition at each
$E_i$ is precisely membership in $\mathcal I_{p_i}^{\,m}$.
But $m=ks>kw$, contradicting the hypothesis. Therefore $D_0$ is nef.
\end{proof}

For completeness, Riemann--Roch and a count of jet conditions also give
the universal upper bound for the Seshadri constant.

\begin{lemma}[The square upper bound]\label{lem:sesha-upper}
For every ample line bundle $L$ on a smooth integral complex projective
surface $S$ and every tuple of $r$ distinct points,
\[
 \varepsilon(S,L;p_1,\ldots,p_r)\le\sqrt{L^2/r}.
\]
\end{lemma}

\begin{proof}
Put $H=L^2$, let $\pi:Y\to S$ be the blow-up, and write
$A=\pi^*L$ and $E=\sum_iE_i$. Suppose $A-\lambda E$ is nef for
some $\lambda\ge0$. Fix $0<\alpha<\sqrt{H/r}$ and set
$m_k=\lceil k\alpha\rceil$. The proof of
Lemma~\ref{lem:positive-square}, applied on $S$ with $A=D=L$,
gives
\[
 h^0(S,L^{\otimes k})\ge \frac{H}{2}k^2+O(k).
\]
The dimension of the target of the order-$m_k$ jet evaluation at
the $r$ points is
\[
 r\binom{m_k+1}{2}=\frac{r\alpha^2}{2}k^2+O(k).
\]
Since $r\alpha^2<H$, evaluation has a nonzero kernel for all
sufficiently large $k$. A section in that kernel gives an effective
divisor $F_k\sim kA-m_kE$ on $Y$. Nefness implies
\[
 0\le(A-\lambda E)\cdot F_k=kH-\lambda r m_k.
\]
Letting $k\to\infty$ gives $\lambda\le H/(r\alpha)$, and then
letting $\alpha\uparrow\sqrt{H/r}$ gives
$\lambda\le\sqrt{H/r}$. Taking the supremum over the nef
coefficients $\lambda$ proves the claim.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:main}]
Choose the nodal divisor and parameters in
Section~\ref{sec:node}. Corollary~\ref{cor:torus-test} supplies an
integer $r_0$, depending only on these choices and hence on $(S,L)$,
with the following property. For every integer $r\ge r_0$, there is
a parameter $t>T$ and an $r$-point torus tuple such that the jet test
on $\mathcal M(kP_t)$ is injective for every pair of positive integers $k,m$
with $m>k\sqrt{H/r}$.

Fix any such integer $r$. For each of these pairs,
Corollary~\ref{cor:surface-transfer} gives an injective jet test on
$H^0(S,L^{\otimes k})$ at some tuple of $r$ distinct points of $S$.
The tuples may depend on $k,m$. Lemma~\ref{lem:jet-open} makes all
these countably many tests hold simultaneously at very general
tuples. Proposition~\ref{prop:homogeneous-nef} proves that the class
$\pi^*L-\sqrt{H/r}\sum_iE_i$ is nef at every such tuple.
Combining this lower bound for the Seshadri constant with
Lemma~\ref{lem:sesha-upper} proves the required equality.
The integer $r$ was arbitrary subject to $r\ge r_0$, and $r_0$
was fixed before $k,m$ were chosen. This proves the assertion for
every sufficiently large integer $r$.
\end{proof}

\subsection{Extension of the ground field}

\begin{corollary}\label{cor:fields}
Let $K$ be an uncountable algebraically closed field of characteristic
zero, let $S$ be a smooth integral projective surface over $K$, and let
$L$ be an ample line bundle on $S$. There is an integer $r_0=r_0(S,L)$
such that, for every $r\ge r_0$, outside a countable union of proper
Zariski-closed subsets of $U_r$, with nonempty complement, one has
\[
 \varepsilon(S,L;p_1,\ldots,p_r)=\sqrt{L^2/r}.
\]
\end{corollary}
\begin{proof}
Descend the projective equations and the finite gluing data for $L$ to
a subfield $F\subset K$ finitely generated over $\mathbb Q$. This gives
a smooth geometrically integral projective surface $S_0/F$ and an ample
line bundle $L_0$ whose base change is $(S,L)$; these properties descend
through field extension, and the finite-data descent and ampleness
criteria are recorded in \cite[Tags 01ZM and 0D3C]{Stacks}.
The field $F$ is countable and embeds into $\C$. Fix such an embedding
and choose $r_0$ from Theorem~\ref{thm:main} for $(S_{0,\C},L_{0,\C})$.
The intersection number $H=L_0^2=L^2$ is unchanged by field extension.

Fix $r\ge r_0$ and put $w=\sqrt{H/r}$. For every pair of positive
integers $a,m$ with $m>aw$, form on $U_{r,0}=U_r(S_0)$ the evaluation map
\[
 H^0(S_0,L_0^{\otimes a})\otimes\mathcal O_{U_{r,0}}
 \longrightarrow
 \bigoplus_{i=1}^r\operatorname{pr}_i^*\mathcal J_{a,m}
\]
using the principal-parts construction in the proof of
Lemma~\ref{lem:jet-open}. Its noninjectivity locus $B_{a,m}$ is closed,
cut out by maximal minors, and is empty if the source space is zero.
Principal parts and these minors commute with field extension, as does
$H^0$ by flat base change \cite[Tag 02KH]{Stacks}.
Each $B_{a,m}$ is proper over $\C$: at a tuple where
$A-wE$ is nef, with $A=\pi^*L_{0,\C}$ and $E=\sum_iE_i$, a section in
the evaluation kernel would give an effective divisor of class $aA-mE$,
whose intersection with $A-wE$ is $aH-rmw<0$.
Thus $B_{a,m}$ is proper over $F$ and remains proper over $K$.
The single threshold $r_0$ works for all these pairs.

Let $Z_r=\bigcup_{a,m:\,m>aw}B_{a,m,K}$. This is a countable union of
proper closed subsets. Its complement contains a $K$-point: the
function field $F(U_{r,0})$ has transcendence degree $2r$ over $F$ and
embeds over $F$ into $K$. Indeed, choose $2r$ algebraically independent
elements of $K$ over the countable field $F$ and extend the resulting
embedding of a rational function field across the finite algebraic
extension, using that $K$ is algebraically closed. The resulting
$K$-point maps to the generic point of $U_{r,0}$ and therefore avoids
all the $B_{a,m}$.

At every tuple outside $Z_r$, all the homogeneous jet tests are
injective. The proofs of Lemma~\ref{lem:positive-square},
Proposition~\ref{prop:homogeneous-nef}, and Lemma~\ref{lem:sesha-upper}
use only surface Riemann--Roch, Serre duality, and the blow-up ideal
identity, and are valid over $K$. They give nefness of
$\pi^*L-w\sum_iE_i$ and the matching upper bound, respectively.
\end{proof}
==== END Maximal-Multipoint-Seshadri-Constants-in-Higher-Dimensions//nef.tex ====

==== BEGIN Maximal-Multipoint-Seshadri-Constants-in-Higher-Dimensions//node.tex ====
\section{A nodal divisor and its exponent diagram}
\label{sec:node}

We now bound the initial exponents of sections of an ample line bundle by
intersecting their divisors with one fixed nodal curve.  Varying the weight
in the two branch coordinates produces a family of quadrilaterals.  The
next subsection will compress these diagrams to triangles.

\begin{lemma}[A nodal ample divisor]
\label{lem:node}
Let $S$ be a smooth integral complex projective surface and let $L$ be an
ample line bundle.  There exist a positive integer $d$, an integral divisor
$V\in|dL|$, and a point $p\in V$ such that $V$ is smooth away from $p$ and
has two smooth transverse analytic branches at $p$.
\end{lemma}

\begin{proof}
Choose an integer $e>0$ such that $eL$ is very ample, embed $S$ in
$\PP^N$ using this line bundle, and choose any $p\in S$.  Take homogeneous
coordinates with $p=[1:0:\cdots:0]$.  The restrictions of $X_1,\ldots,X_N$
generate $\mathcal I_p\otimes\mathcal O_S(eL)$: away from $p$ at least one
is nonzero, and near $p$ their ratios with $X_0$ generate the maximal ideal
because the embedding is a closed immersion.

Let $\pi:X=\operatorname{Bl}_pS\to S$ be the blow-up, with exceptional
curve $E$.  The generators just described give a surjection onto the
twisted Rees algebra
\[
 \operatorname{Sym}^{\bullet}(\mathcal O_S^{\oplus N})
 \longrightarrow
 \bigoplus_{n\ge0}\mathcal I_p^n\otimes\mathcal O_S(neL).
\]
Taking relative $\operatorname{Proj}$ gives a closed immersion
$X\hookrightarrow S\times\PP^{N-1}$, the closure of the graph of projection from $p$.
The second factor pulls $\mathcal O(1)$ back to
$\mathcal O_X(e\pi^*L-E)$; see also \cite[Tag 01OF]{Stacks}.
After embedding the first factor in $\PP^N$, the very ample bundle
$\mathcal O(1,1)$ therefore restricts to
\[
 \mathcal A=\mathcal O_X(2e\pi^*L-E).
\]
In particular, $\mathcal A^{\otimes2}=\mathcal O_X(4e\pi^*L-2E)$ is
very ample.

The point blow-up $X$ is a smooth integral projective surface.  By
Bertini, a general divisor $\widetilde V$ in this last linear system
is smooth and integral \cite[Tags 0FD6 and 0G4F]{Stacks}.  We may also
require that it meet $E$ transversely at two distinct points.  Indeed,
$\mathcal A|_E\simeq\mathcal O_{\PP^1}(1)$, and the restrictions of its
global sections generate this bundle, hence span its full two-dimensional
space of sections.  Products of these sections show that
\[
 H^0(X,\mathcal A^{\otimes2})
 \longrightarrow H^0(E,\mathcal O_{\PP^1}(2))
\]
is surjective.  Having two distinct zeros on $E$ is consequently a
nonempty open condition, which can be imposed along with the Bertini
conditions.

Set $d=4e$ and $V=\pi(\widetilde V)$.  Pushing down the divisor class
shows that $V\in|dL|$, and $V$ is integral because $\widetilde V$ is.
The blow-down is an isomorphism away from $E$, so $V$ is smooth away
from $p$.  In a local blow-up chart the map has the form
$(a_0,u_0)\mapsto(a_0,a_0u_0)$ with $E=\{a_0=0\}$.
A curve transverse to $E$ has $a_0$ as a local parameter, and its image
is a smooth branch.  The two different points of $\widetilde V\cap E$
give different tangent directions.  Thus the two image branches are
smooth and transverse at $p$.
\end{proof}

Fix $d,V,p$ as in Lemma~\ref{lem:node}, and set $H=L^2$.
Choose holomorphic coordinates $(x,z)$ at $p$ in which the two branches
are the coordinate axes, and choose a nonvanishing local frame of $L$.
In this frame a defining section $\tau\in H^0(S,dL)$ of $V$ has the form
\begin{equation}
 \tau=u(x,z)xz,\qquad u(0,0)\ne0.
 \label{eq:node-unit}
\end{equation}
Here and below a section is identified with its local function in the
chosen frame and its tensor powers.

Choose once and for all $T>1$ so large that
\begin{equation}
 (1+t)^2>d^2Ht\qquad\text{for every }t>T.
 \label{eq:weight-range}
\end{equation}
Such a choice is possible because the left side is quadratic in $t$.
For $t>T$, put
\begin{equation}
 \begin{gathered}
 c=\frac1d,\qquad W=dH,\qquad
 a=\frac{Wt}{1+t},\qquad b=\frac{W}{1+t},\\
 Q_t=\conv\{(0,0),(a,0),(c,c),(0,b)\}.
 \end{gathered}
 \label{eq:quadrilateral}
\end{equation}
Recall that $\nu_t(f)$ denotes the first nonzero Taylor exponent of $f$
when exponents $(\alpha,\beta)$ are ordered by
$(\alpha+t\beta,\beta)$.  The inequality \eqref{eq:weight-range} is
equivalent to $c/a+c/b>1$.  Thus $(c,c)$ lies beyond the side joining
$(a,0)$ and $(0,b)$, and the four vertices in
\eqref{eq:quadrilateral} occur in the displayed cyclic order.

\begin{proposition}[The quadrilateral bound]
\label{prop:quadrilateral}
For the surface, line bundle, nodal divisor, coordinates and frame fixed
above, every integer $k\ge1$, every $t>T$, and every nonzero
$s\in H^0(S,kL)$ satisfy
\[
 \nu_t(s)\in kQ_t.
\]
Moreover, $H^0(S,kL)$ has a basis with pairwise distinct initial exponents
in $kQ_t$; the basis is empty if this section space is zero.
\end{proposition}

\begin{proof}
First suppose that the divisor $D$ of $s$ does not contain $V$, and write
$f(x,z)$ for its local function.  Neither $f(x,0)$ nor $f(0,z)$ vanishes
identically.  To see this, $D\cap V$ is a proper closed subset of the
integral algebraic curve $V$, hence is finite.  An identically vanishing
branch restriction would put an entire analytic branch in this finite
set.  This argument uses the global integrality of $V$ without asserting
that its nodal analytic germ is irreducible.

Let $\ell_x=\ord_0f(x,0)$ and $\ell_z=\ord_0f(0,z)$, and let $v$ be
the weight of $\nu_t(f)$.  The pure terms defining these two finite
orders have weights $\ell_x$ and $t\ell_z$, respectively.  Therefore
$\ell_x\ge v$ and $\ell_z\ge v/t$.  Intersection multiplicity is additive
over the two smooth branches, so
\begin{equation}
 v\left(1+\frac1t\right)
 \le \ell_x+\ell_z
 = I_p(D,V)
 \le D\cdot V=kdH.
 \label{eq:branch-bound}
\end{equation}
The analytic computation gives the algebraic local intersection number:
both are the colength of $(f,xz)$ in the common completed local ring
$\C[[x,z]]$.  The final inequality in \eqref{eq:branch-bound} follows
from nonnegativity of all the other local intersection numbers.
Consequently $v\le ka$, and
\begin{equation}
 \nu_t(s)\in k\Delta_t,\qquad
 \Delta_t=\conv\{(0,0),(a,0),(0,b)\},
 \label{eq:residual-triangle}
\end{equation}
since $a=tb$.

For a general nonzero $s$, let $j$ be the coefficient of $V$ in its
divisor.  Dividing by the defining section of this effective Cartier
divisor gives
\[
 s=\tau^j\rho,\qquad
 \rho\in H^0(S,nL),\qquad n=k-jd,
\]
where the effective divisor $R$ of $\rho$ does not contain $V$.
Intersecting with the ample line bundle gives
$nH=L\cdot R\ge0$, so $n\ge0$.  If $n=0$, then $R=0$ by ampleness
and $\rho$ is a nonzero constant, since $S$ is integral and projective.
If $n>0$, the preceding argument gives $\nu_t(\rho)\in n\Delta_t$.

The order on exponents is compatible with addition, so initial exponents
add under multiplication.  A unit has initial exponent $(0,0)$.
Equation~\eqref{eq:node-unit} therefore yields
\[
 \nu_t(s)=(j,j)+\nu_t(\rho).
\]
When $n>0$, this implies
\[
 \frac{\nu_t(s)}{k}
 =\frac{n}{k}\,\frac{\nu_t(\rho)}{n}
  +\frac{jd}{k}(c,c)\in Q_t,
\]
because the two coefficients are nonnegative and sum to one.  When
$n=0$, the normalized exponent equals $(c,c)$ directly.

Finally, restriction to the analytic chart is injective on global
sections, by the identity theorem.  Apply the finite coefficient
elimination in Proposition~\ref{prop:initial-transfer} to this
finite-dimensional space of germs.  It gives a basis with distinct
initial exponents, each of which satisfies the bound just proved.
\end{proof}

Proposition~\ref{prop:quadrilateral} controls the initial exponents; it
makes no claim about the remaining Taylor coefficients.  We next use
slices parallel to $(1,1)$ to replace its quadrilateral by a triangle
for the purpose of testing jets.
==== END Maximal-Multipoint-Seshadri-Constants-in-Higher-Dimensions//node.tex ====

==== BEGIN Maximal-Multipoint-Seshadri-Constants-in-Higher-Dimensions//preamble.tex ====
\documentclass[11pt]{article}
\pdftrailerid{}
\usepackage[T1]{fontenc}
\usepackage{lmodern}
\usepackage[margin=1.05in]{geometry}
\usepackage{amsmath,amssymb,amsthm,mathtools,microtype}
\usepackage{enumitem,xcolor,tikz}
\usetikzlibrary{arrows.meta,calc,positioning,patterns}
\usepackage[colorlinks=true,linkcolor=blue!45!black,citecolor=blue!45!black,urlcolor=blue!45!black]{hyperref}
\setlength{\emergencystretch}{2em}
\setlist{itemsep=3pt,topsep=5pt}
\newcommand{\C}{\mathbb C}
\newcommand{\R}{\mathbb R}
\newcommand{\Z}{\mathbb Z}
\newcommand{\N}{\mathbb N}
\newcommand{\T}{(\C^\times)^2}
\newcommand{\PP}{\mathbb P}
\DeclareMathOperator{\conv}{conv}
\DeclareMathOperator{\ord}{ord}
\DeclareMathOperator{\mult}{mult}
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\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
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\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}
\numberwithin{equation}{section}
\hypersetup{pdftitle={Maximal Seshadri constants on arbitrary polarized surfaces},pdfauthor={OpenAI}}
==== END Maximal-Multipoint-Seshadri-Constants-in-Higher-Dimensions//preamble.tex ====

==== BEGIN Maximal-Multipoint-Seshadri-Constants-in-Higher-Dimensions//provenance/companion/main.tex ====
\input{preamble}
\title{Maximal Seshadri constants on arbitrary polarized surfaces}
\author{OpenAI}
\date{September 23, 2026}
\begin{document}
\maketitle
\begin{abstract}
We prove that, for every smooth integral complex projective surface $S$
and every ample line bundle $L$, the multipoint Seshadri constant at $r$
very general points equals $\sqrt{L^2/r}$ for every sufficiently large
integer $r$. This resolves positively the qualitative Nagata--Biran
conjecture for surfaces. The proof combines finite jet interpolation near
a nodal curve with torus subgroups of prescribed order.
\end{abstract}
\tableofcontents
\input{sections/introduction}
\input{sections/jets}
\input{sections/node}
\input{sections/compression}
\input{sections/lattice}
\input{sections/nef}
\bibliographystyle{alpha}
\begingroup
\small
\bibliography{references}
\endgroup
\end{document}
==== END Maximal-Multipoint-Seshadri-Constants-in-Higher-Dimensions//provenance/companion/main.tex ====

==== BEGIN Maximal-Multipoint-Seshadri-Constants-in-Higher-Dimensions//provenance/companion/sections/compression.tex ====
\subsection{Compressing diagonal slices}\label{sec:compression}

The quadrilateral in Proposition~\ref{prop:quadrilateral} bounds the initial
exponents of sections.  We next reduce its jet tests to those of the triangle
\[
 P_t=\conv\{(-b,0),(a,0),(0,c)\}.
\]
Since $a+b=dH$ and $c=1/d$, this triangle has area
$c(a+b)/2=H/2$.
The reduction preserves the length of each diagonal slice and moves its
lower endpoint to height zero.  Throughout this subsection we test the full
monomial spaces $\mathcal M(kQ_t)$ and $\mathcal M(kP_t)$.  The initial
monomials of sections need only span a subspace of the former.

\begin{lemma}[Diagonal slices]\label{lem:slices}
For $-b\le q\le a$, the slice of $Q_t$ on the line
$\alpha-\beta=q$ has length in the $\beta$ coordinate equal to
\[
 \ell_t(q)=
 \begin{cases}
 c(1+q/b),&-b\le q\le0,\\[2pt]
 c(1-q/a),&0\le q\le a.
 \end{cases}
\]
There are no slices outside this interval, and
\[
 P_t=\{(q,h):-b\le q\le a,\ 0\le h\le\ell_t(q)\}.
\]
\end{lemma}

\begin{proof}
The strict inequality defining $t$ is equivalent to
\[
 c(a+b)>ab.
\]
Thus $(c,c)$ lies strictly beyond the segment joining $(a,0)$ to $(0,b)$,
on the side away from the origin.  In the coordinates
$(q,\beta)=(\alpha-\beta,\beta)$, the four vertices of $Q_t$, in cyclic
order, are
\[
 (0,0),\quad (a,0),\quad (0,c),\quad (-b,b).
\]
For $0\le q\le a$, the slice runs from $\beta=0$ to
$\beta=c(1-q/a)$.  For $-b\le q\le0$, it runs from
\[
 \beta=-q
 \qquad\text{to}\qquad
 \beta=c+\frac{c-b}{b}q.
\]
Subtracting the endpoints gives the stated formula.  The region below
this tent is precisely $P_t$, as illustrated in Figure~\ref{fig:compression}.
\end{proof}

\input{figures/compression}


\begin{proposition}[Diagonal compression]\label{prop:compression}
Let $k,m,r$ be positive integers.  If $\mathcal M(kP_t)$ has injective
order-$m$ jets at some $r$ distinct points of $\T$, then
$\mathcal M(kQ_t)$ has injective order-$m$ jets at some $r$ distinct
points of $\T$.
\end{proposition}

\begin{proof}
Use the torus coordinates
\[
 (x,z)=(u,v/u),\qquad (u,v)=(x,xz).
\]
A monomial $x^\alpha z^\beta$ becomes $u^qv^\beta$, where
$q=\alpha-\beta$.  For every nonempty integer slice $q$ of $kQ_t$, write
\[
 \beta_q^-=
 \min\{\beta\in\Z:(q+\beta,\beta)\in kQ_t\},
 \qquad
 \beta_q^+=
 \max\{\beta\in\Z:(q+\beta,\beta)\in kQ_t\}.
\]
Convexity of the slice implies that all integers between these endpoints
occur.  Hence the slice has the basis
\begin{equation}\label{eq:compression-row-basis}
 u^qv^{\beta_q^-}(v-1)^h,
 \qquad 0\le h\le\beta_q^+-\beta_q^-.
\end{equation}
Indeed, expanding $(v-1)^h$ gives a triangular change of basis from the
consecutive powers of $v$, with diagonal entries equal to one.  By Lemma~\ref{lem:slices},
\[
 h\le\beta_q^+-\beta_q^-
 \le k\ell_t(q/k).
\]
Consequently every pair $(q,h)$ in
\eqref{eq:compression-row-basis} belongs to $kP_t\cap\Z^2$.
These pairs are distinct, both within each slice and between different slices.

Let $(u_i,y_i)$, $1\le i\le r$, be distinct torus points giving an
injective jet test for $\mathcal M(kP_t)$, now written in the variables
$(u,y)$.  For a positive parameter $s$, substitute $v=1+sy$ into
\eqref{eq:compression-row-basis} and divide its member indexed by $h$
by $s^h$.  The resulting functions are
\begin{equation}\label{eq:compression-limit}
 u^q(1+sy)^{\beta_q^-}y^h
 \longrightarrow u^qy^h
 \qquad (s\longrightarrow0^+).
\end{equation}
The convergence is locally uniform, with every spatial derivative, on
fixed neighborhoods of the finitely many test points.  To see this,
choose bounded neighborhoods on which $u$ stays away from zero and
then take $s$ small enough that $|sy|<1/2$ there.  The factors
$(1+sy)^{\beta_q^-}$ are holomorphic and converge uniformly to one on
slightly larger neighborhoods, so the derivative assertion follows
from the Cauchy estimates.  The argument also applies to negative
Laurent powers, since $1+sy$ stays away from zero.

The distinct limiting monomials in \eqref{eq:compression-limit} span
a subspace of $\mathcal M(kP_t)$, so their jet matrix has full column
rank.  Lemma~\ref{lem:rank-persistence} gives the same conclusion for
the functions on the left for every sufficiently small positive $s$.
These functions form a rescaled pullback of a basis of
$\mathcal M(kQ_t)$.

For such an $s$, the corresponding points in the original torus are
\[
 \left(u_i,\frac{1+sy_i}{u_i}\right),\qquad 1\le i\le r.
\]
They lie in $\T$ because $u_i\ne0$ and $1+sy_i\ne0$.  They are distinct:
equality of two images forces equality first of their $u$ coordinates
and then of their $y$ coordinates.  The coordinate map
\[
 (u,y)\longmapsto\left(u,\frac{1+sy}{u}\right)
\]
has Jacobian determinant $s/u\ne0$, and therefore preserves orders of
vanishing at these points.  The full-column-rank conclusion is thus
exactly the required injectivity for $\mathcal M(kQ_t)$.
\end{proof}

The two transfers now give the test needed on the surface.

\begin{corollary}[From the triangle to the surface]
\label{cor:surface-transfer}
Let $k,m,r$ be positive integers.  If $\mathcal M(kP_t)$ has injective
order-$m$ jets at some $r$ distinct torus points, then $H^0(S,kL)$ has
injective order-$m$ jets at some $r$ distinct points of $S$.
\end{corollary}

\begin{proof}
If $H^0(S,kL)=0$, the conclusion is immediate.  Otherwise,
Proposition~\ref{prop:compression} gives such a jet test for
$\mathcal M(kQ_t)$.  Trivialize $L$ in the fixed coordinate chart at
the node and regard the sections of $kL$ as holomorphic germs there.
This identification is injective: a global section vanishing on a
nonempty open subset of the integral surface vanishes identically.
Proposition~\ref{prop:quadrilateral} gives a basis whose distinct
initial monomials lie in $kQ_t$.  Their span therefore has injective
jets at the torus tuple just constructed.  Applying
Proposition~\ref{prop:initial-transfer} gives injective jets for the
section germs at distinct points in the chart, and hence for
$H^0(S,kL)$ on $S$.
\end{proof}
==== END Maximal-Multipoint-Seshadri-Constants-in-Higher-Dimensions//provenance/companion/sections/compression.tex ====

==== BEGIN Maximal-Multipoint-Seshadri-Constants-in-Higher-Dimensions//reshaping.tex ====
\newcommand{\ReshapingNodeFigure}{%
\begin{figure}[H]
\centering
\begin{tikzpicture}[x=.88cm,y=2.8cm,font=\small,
  polyshape/.style={draw=teal!65!black,fill=teal!10,line width=.8pt},
  slice/.style={draw=orange!85!black,line width=1.2pt}]
  \begin{scope}
    \node at (1.4,.77) {$Q$};
    \draw[->,gray!65] (-.2,0)--(3.35,0);
    \draw[->,gray!65] (0,-.04)--(0,.63);
    \path[polyshape] (0,0)--(3,0)--(.5,.5)--(0,.375)--cycle;
    \node[below left] at (0,0) {$0$};
    \node[below] at (3,0) {$(a,0)$};
    \node[above right] at (.5,.5) {$(c,c)$};
    \node[left] at (0,.375) {$(0,b)$};
    \draw[slice] (0,.18)--(.26,.44);
    \node at (1.4,-.21) {diagonal slices};
  \end{scope}
  \begin{scope}[xshift=4.65cm]
    \node at (1.35,.77) {after the shear};
    \draw[->,gray!65] (-.7,0)--(3.35,0);
    \draw[->,gray!65] (0,-.04)--(0,.63);
    \path[polyshape] (0,0)--(3,0)--(0,.5)--(-.375,.375)--cycle;
    \node[below right] at (0,0) {$0$};
    \node[below] at (3,0) {$(a,0)$};
    \node[above right] at (0,.5) {$(0,c)$};
    \node[left] at (-.375,.375) {$(-b,b)$};
    \draw[slice] (-.18,.18)--(-.18,.44);
    \node at (1.35,-.21) {vertical slices};
  \end{scope}
  \begin{scope}[xshift=9.3cm]
    \node at (1.35,.77) {$P$};
    \draw[->,gray!65] (-.7,0)--(3.35,0);
    \draw[->,gray!65] (0,-.04)--(0,.63);
    \path[polyshape] (-.375,0)--(3,0)--(0,.5)--cycle;
    \node[below left] at (-.375,0) {$(-b,0)$};
    \node[below] at (3,0) {$(a,0)$};
    \node[above right] at (0,.5) {$(0,c)$};
    \draw[slice] (-.18,0)--(-.18,.26);
    \node at (1.35,-.21) {slices moved to height $0$};
  \end{scope}
  \draw[-{Stealth[length=2mm]},thick] (3.65,.23)--(4.25,.23);
  \draw[-{Stealth[length=2mm]},thick] (8.9,.23)--(9.5,.23);
\end{tikzpicture}
\caption{Bounding polygons in the diagonal compression.  The shear turns
lines of direction $(1,1)$ into vertical lines.  Moving each vertical slice
to start at height zero produces $P$; the marked slices correspond under
these operations.  These are continuous bounds: actual row exponents are
replaced by possible vanishing orders, not translated point by point.
All three polygons have area $H/2$.  Axis scales are chosen for legibility.}
\label{fig:node-compression}
\end{figure}%
}

\newcommand{\ReshapingCompressionFigure}{%
\begin{figure}[H]
\centering
\begin{tikzpicture}[x=.88cm,y=.88cm,font=\small,
  polyshape/.style={draw=teal!65!black,fill=teal!10,line width=.8pt},
  slice/.style={draw=orange!85!black,line width=1.2pt}]
  \begin{scope}
    \node at (1.5,3.25) {in the basis $(u,u')$};
    \path[polyshape] (1,0)--(3,.9)--(.4,2.4)--cycle;
    \draw[densely dashed,gray] (-.1,0)--(1,0);
    \draw[densely dashed,gray] (-.1,.9)--(.775,.9);
    \draw[densely dashed,gray] (-.1,2.4)--(.4,2.4);
    \node[left] at (-.1,0) {$q_0$};
    \node[left] at (-.1,.9) {$q_1$};
    \node[left] at (-.1,2.4) {$q_2$};
    \draw[slice] (.775,.9)--(3,.9);
    \node[above] at (1.9,.9) {$w$};
    \node at (1.5,-.62) {height range $R$};
  \end{scope}
  \begin{scope}[xshift=4.65cm]
    \node at (1.35,3.25) {horizontal compression};
    \draw[->,gray!65] (0,-.12)--(0,2.75);
    \path[polyshape] (0,0)--(2.225,.9)--(0,2.4)--cycle;
    \node[left] at (0,0) {$q_0$};
    \node[left] at (0,.9) {$q_1$};
    \node[left] at (0,2.4) {$q_2$};
    \draw[slice] (0,.9)--(2.225,.9);
    \node[right] at (2.225,.9) {$(w,q_1)$};
    \draw[gray!70,densely dashed] (.85,.344)--(.85,1.827);
    \node at (1.35,-.62) {maximum slice length $w$};
  \end{scope}
  \begin{scope}[xshift=9.3cm]
    \node at (1.4,3.25) {vertical compression};
    \draw[->,gray!65] (-.12,0)--(2.65,0);
    \draw[->,gray!65] (0,-.12)--(0,2.75);
    \path[polyshape] (0,0)--(2.225,0)--(0,2.4)--cycle;
    \node[below left] at (0,0) {$0$};
    \node[below] at (2.225,0) {$w$};
    \node[left] at (0,2.4) {$R$};
    \draw[slice] (.85,0)--(.85,1.483);
    \node at (1.4,-.62) {$\Delta(w,R)$};
  \end{scope}
  \draw[-{Stealth[length=2mm]},thick] (3.55,1.65)--(4.15,1.65);
  \draw[-{Stealth[length=2mm]},thick] (8.75,1.65)--(9.35,1.65);
\end{tikzpicture}
\caption{The primitive direction is chosen so that the transverse height
range is $R=H/w$.  Since the area is $H/2$, the longest horizontal slice
has length $w$.  Horizontal compression aligns the left endpoints of
these slices.  Vertical compression then gives the right triangle with
intercepts $w$ and $R$.  The middle and right panels mark vertical slices
of equal length; the drawing is schematic.}
\label{fig:final-compressions}
\end{figure}%
}
==== END Maximal-Multipoint-Seshadri-Constants-in-Higher-Dimensions//reshaping.tex ====

==== BEGIN Maximal-Multipoint-Seshadri-Constants-in-Higher-Dimensions//saturation.tex ====
\section{Interior lattice points and interpolation}
\label{sec:saturation}

The reshaping proposition preserves the number of monomials, but does not
describe each resulting exponent. For interpolation we need more than the
correct asymptotic number: we need specific sets of exponents. Additivity
across degrees supplies the missing information. An additive family $F_k$
occupying asymptotically the full volume of $kP$ contains every lattice point
of $kK$ for each fixed compact set $K\subset\operatorname{int}P$, once the
degree $k$ is sufficiently large.

The general semigroup approximation theorem of
\citet[Theorem~1.6]{KavehKhovanskii2012} provides a broader setting for
interior-saturation statements. Under the full-density hypothesis needed
here, the following direct proof uses only decompositions into two summands.

\begin{lemma}[Interior saturation]
\label{lem:interior-saturation}
Let $n\ge1$ be an integer, let $P\subset\R^n$ be a compact convex set with
nonempty interior, and let $F_k\subset kP\cap\Z^n$ for every integer
$k\ge1$. Suppose that
\[
 F_k+F_l\subset F_{k+l}\quad(k,l\ge1),
 \qquad
 \#F_k=\vol(P)k^n+o(k^n).
\]
For every compact set $K\subset\operatorname{int}P$, there is an integer
$k_K$ such that
\[
 kK\cap\Z^n\subset F_k\qquad(k\ge k_K).
\]
\end{lemma}

\begin{proof}
The lattice-point estimate for a full-dimensional compact convex set gives
\[
 \#(kP\cap\Z^n)=\vol(P)k^n+o(k^n).
\]
Consequently the sets of missing exponents
\[
 H_k=(kP\cap\Z^n)\setminus F_k
\]
satisfy $\#H_k=o(k^n)$. This estimate alone would allow individual missing
points deep inside $kP$. We use additivity to exclude them.

We may assume that $K$ is nonempty. Choose $\delta>0$ so that every closed
ball of radius $\delta$ centered in $K$ is contained in $P$. For a large
integer $k$, put $k_1=\lfloor k/2\rfloor$ and $k_2=k-k_1$, and fix
$y\in kK\cap\Z^n$. Write $z=y/k$. Every integer vector $y_1$ satisfying
\[
 \lVert y_1-k_1z\rVert\le\delta\min(k_1,k_2)
\]
gives a decomposition $y=y_1+y_2$ with $y_i\in k_iP\cap\Z^n$. Indeed,
both $y_1/k_1$ and $y_2/k_2$ are within distance $\delta$ of $z$.
The ball in this display contains at least $c k^n$ lattice points for all
sufficiently large $k$, where $c>0$ is independent of $y$.

At most $\#H_{k_1}$ choices have $y_1\notin F_{k_1}$, and at most
$\#H_{k_2}$ choices have $y_2\notin F_{k_2}$, because $y_1\mapsto y-y_1$
is injective. The total number excluded is $o(k^n)$. Thus some decomposition
has $y_i\in F_{k_i}$, and additivity gives $y\in F_k$. All constants and
degree thresholds used here are uniform over $y\in kK\cap\Z^n$.
\end{proof}

The lemma requires no finite-generation hypothesis on the sets $F_k$ or on
the semigroup they form. It also permits real, rather than rational,
bounding polytopes. We next describe the exponents needed for interpolation
in the simplex that will result from repeated reshaping.

\begin{lemma}[Interpolation in an elongated simplex]
\label{lem:simplex-interpolation}
Let $n\ge2$, $r\ge1$, and $m\ge0$ be integers, and let
$c_1,\ldots,c_r\in\C^*$ be distinct. Put
\[
 E_{r,m}=
 \left\{(\beta,b)\in\Z_{\ge0}^{n-1}\times\Z_{\ge0}:
       |\beta|+\frac br<m+1\right\}.
\]
Then $\M(E_{r,m})$ surjects onto jets through degree $m$ at the points
\[
 q_i=(1,\ldots,1,c_i)\in\Torus^n,\qquad 1\le i\le r.
\]
\end{lemma}

\begin{proof}
Write the coordinates as $(x',Y)$, with $n-1$ coordinates in $x'$. At $q_i$,
an arbitrary jet through degree $m$ has a unique expression
\[
 \sum_{|\alpha|\le m}(x'-1)^\alpha h_{i,\alpha}(Y-c_i),
 \qquad
 \deg h_{i,\alpha}\le m-|\alpha|.
\]
For each $\alpha$, the Chinese remainder theorem provides a polynomial
$f_\alpha(Y)$ such that
\[
 \begin{split}
 f_\alpha(Y)&\equiv h_{i,\alpha}(Y-c_i)
       \pmod{(Y-c_i)^{m+1-|\alpha|}}\quad(1\le i\le r),\\
 \deg f_\alpha&<r(m+1-|\alpha|).
 \end{split}
\]
The polynomial
\[
 f(x',Y)=\sum_{|\alpha|\le m}(x'-1)^\alpha f_\alpha(Y)
\]
has all the prescribed jets. Every monomial $(x')^\beta Y^b$ occurring in
its $\alpha$-summand satisfies $\beta\le\alpha$ componentwise and hence
\[
 |\beta|+\frac br
 \le |\alpha|+\frac br
 <|\alpha|+(m+1-|\alpha|)=m+1.
\]
Thus $f\in\M(E_{r,m})$.
\end{proof}

The two lemmas now convert asymptotically full additive supports into
simultaneous jet surjectivity. Translating a slightly smaller simplex into
the interior avoids any assertion about exponents on the boundary.

\begin{proposition}[Jets from full-volume supports]
\label{prop:asymptotic-jets}
Let $n\ge2$ and $r\ge1$ be integers, let $w>0$, and set
\[
 P=\Delta(w,\ldots,w,rw).
\]
Suppose that finite sets $F_k\subset kP\cap\Z^n$, indexed by all integers
$k\ge1$, satisfy
\[
 F_k+F_l\subset F_{k+l},
 \qquad
 \#F_k=\vol(P)k^n+o(k^n).
\]
For every $0<\theta<w$ and every choice of distinct
$c_1,\ldots,c_r\in\C^*$, the space $\M(F_k)$ surjects onto jets through
degree $\lfloor k\theta\rfloor$ at $(1,\ldots,1,c_i)$ for all sufficiently
large $k$.
\end{proposition}

\begin{proof}
Choose $\theta/w<\lambda<1$. There is a vector $a$ with strictly positive
coordinates such that $a+\lambda P\subset\operatorname{int}P$: it suffices
to require
\[
 \frac{a_1+\cdots+a_{n-1}}w+\frac{a_n}{rw}<1-\lambda.
\]
Choose $a_k\in k^{-1}\Z^n$ with $a_k\to a$. For all sufficiently large
$k$, the translated simplices $a_k+\lambda P$ lie in a single compact
subset $K$ of $\operatorname{int}P$. Lemma~\ref{lem:interior-saturation}
therefore gives
\[
 ka_k+(k\lambda P\cap\Z^n)\subset F_k.
\]
Consequently $\M(F_k)$ contains the monomial space for
$k\lambda P\cap\Z^n$, multiplied by the monomial with exponent $ka_k$.
Multiplication by this monomial is an isomorphism on each jet algebra at
a torus point, since the monomial is a unit there.

Put $m=\lfloor k\theta\rfloor$. For all sufficiently large $k$,
$m+1\le k\lambda w$, and hence
\[
 E_{r,m}\subset k\lambda P\cap\Z^n.
\]
Lemma~\ref{lem:simplex-interpolation} supplies jet surjectivity for this
subspace. The monomial translation preserves surjectivity and proves the
claim.
\end{proof}
==== END Maximal-Multipoint-Seshadri-Constants-in-Higher-Dimensions//saturation.tex ====

==== BEGIN Maximal-Multipoint-Seshadri-Constants-in-Higher-Dimensions//sections/introduction.tex ====
\section{Introduction}\label{sec:introduction}

Let $X$ be a smooth integral complex projective variety of dimension $n\ge3$,
and let $L$ be an ample line bundle. Put $V=L^n$. For a tuple
$\mathbf p=(p_1,\ldots,p_r)$ of distinct points, the ordinary multipoint
Seshadri constant is
\begin{equation}\label{eq:seshadri-definition}
 \eps(X,L;\mathbf p)
 =\inf_C\frac{L\cdot C}{\sum_{i=1}^r\mult_{p_i}C},
\end{equation}
where $C$ ranges over integral curves meeting at least one of the marked
points, and the multiplicity is zero at points outside $C$. This invariant
measures the local positivity of $L$ at the entire tuple. Its universal
upper bound is
\begin{equation}\label{eq:volume-bound}
 \eps(X,L;\mathbf p)\le (V/r)^{1/n}.
\end{equation}
Equality means that no curve imposes a stronger constraint than the volume
of $L$.

Write
\[
 U_r(X)=\{(p_1,\ldots,p_r)\in X^r:p_i\ne p_j\text{ when }i\ne j\}.
\]
A property holds at \emph{very general} tuples if the exceptional tuples
are contained in a countable union of proper Zariski-closed subsets of
$U_r(X)$.

\begin{theorem}\label{thm:main}
Let $X$ be a smooth integral complex projective variety of dimension
$n\ge3$, and let $L$ be an ample line bundle on $X$. There is an integer
$r_0=r_0(X,L)\ge1$ such that for every integer $r\ge r_0$ there is a
countable union $Z_r$ of proper Zariski-closed subsets of $U_r(X)$, with
nonempty complement, for which
\[
 \eps(X,L;\mathbf p)=\left(\frac{L^n}{r}\right)^{1/n}
 \qquad\text{for every }\mathbf p\in U_r(X)\setminus Z_r.
\]
\end{theorem}

\begin{samepage}
Equivalently, on the blow-up $\pi:Y\to X$ at such a tuple, with exceptional
divisors $E_1,\ldots,E_r$, the real divisor class
\[
 \pi^*L-(V/r)^{1/n}\sum_{i=1}^rE_i
\]
is nef and has top self-intersection zero. Here nefness means nonnegative
intersection with every integral curve. The threshold depends on the fixed
polarized variety, but all integers beyond it are included. In particular,
Theorem~\ref{thm:main} asserts exact equality for each such point count,
rather than convergence of a normalized constant as $r$ increases.
\end{samepage}

\subsection{Background and relation to earlier work}

The plane problem originates in Nagata's work on Hilbert's fourteenth
problem. He conjectured a lower bound on the degree of a plane curve in
terms of its multiplicities at very general points, and proved the strict
inequality when the number of points is a square at least $16$
\cite{Nagata1959}. Multipoint Seshadri constants express the corresponding
positivity question on an arbitrary polarized variety. The qualitative
Nagata--Biran--Szemberg formulation asks for maximality at every sufficiently
large point count. Ro\'e and Ross state this formulation in arbitrary
dimension and distinguish the ordinary curve-defined constant from its
higher-cycle analogues \cite[Conjecture~1.3 and Remark~1.4]{RoeRoss2009}.
We use the ordinary constant throughout, even when $\dim X>2$.

Biran's packing stability theorem supplies an important symplectic
precedent: sufficiently many equal balls can fill a closed symplectic
four-manifold with rational symplectic class with arbitrarily small volume
deficit \cite{Biran1999}; see also \cite[Theorem~6.3]{Biran2001}. The algebraic
question fixes the complex variety and asks for a nef class on its point
blow-ups, so symplectic packing stability is not by itself the required
algebraic statement. In arbitrary dimension, K\"uchle obtained lower bounds
for multipoint Seshadri constants that are asymptotically optimal as the
number of points tends to infinity \cite[Theorem~1.1]{Kuechle1996}.
Biran's product inequality, extended to higher-cycle constants by
Ro\'e--Ross, relates point counts on a variety to point counts on projective
space \cite[Theorem~1.1 and Remark~1.2]{RoeRoss2009}.
These results illuminate the volume scale in \eqref{eq:volume-bound}; an
asymptotically optimal estimate alone does not imply eventual equality.

The companion paper \cite{OpenAI2026Surfaces} establishes the qualitative
surface statement by a nodal exponent bound, diagonal slice compression,
and a primitive lattice direction tuned to the number of points. Its final
tests are \emph{injective} jet evaluations on finite torus subgroups.
We adapt the planar geometric mechanism to polynomial spaces in two
variables and use \emph{surjective} full-dimensional jet evaluations instead.
All arguments required for this adaptation are proved here; the surface
maximality theorem is not an input.

The interpretation of positivity through asymptotic jet generation is
classical; see Demailly \cite[Section~6]{Demailly1992}.
Our use of initial monomials belongs to the valuation approach to linear
series initiated by Okounkov \cite{Okounkov1996} and developed by Lazarsfeld--Musta\c{t}\u{a}
\cite{LazarsfeldMustata2009} and Kaveh--Khovanskii
\cite{KavehKhovanskii2012}. More directly, Ito compares multipoint jet
separation with initial monomial spaces by separate finite-degree
degenerations, without a finite-generation hypothesis
\cite[Proposition~5.7 and Theorem~5.8]{Ito2013}.
Monomial diagrams and interpolation minors also underlie Dumnicki's
diagram-cutting method \cite[Proposition~13 and Theorem~14]{Dumnicki2007}.
We retain elementary proofs of the specific transfer and counting statements
used below.

\subsection{Proof strategy and the two-coordinate construction}

Separating jets through degree $m$ at a tuple means prescribing,
independently at each point, all Taylor coefficients of total degree at
most $m$, in local coordinates and line-bundle frames. If $H^0(X,kL)$
separates these jets, then
\[
 \eps(X,L;\mathbf p)\ge \frac{m}{k};
\]
the curve-intersection argument is given in Lemma~\ref{lem:jets-curves}.
Thus, for $w=(V/r)^{1/n}$, our interpolation target is jet separation
through degree $\lfloor k\theta\rfloor$ for every fixed $0<\theta<w$
and all sufficiently large $k$. We first construct a tuple for each such
test and later obtain simultaneous tests at very general tuples.

The argument works with finite exponent sets
$S_k\subset\Z^n$, one for each positive integer degree $k$, satisfying
\[
 S_k+S_l\subset S_{k+l}.
\]
For a set $S$, its monomial space is the span of the Laurent monomials
$z^\alpha$ with $\alpha\in S$ on $(\C^*)^n$. The multiplication condition
will be as important as the geometry of the individual sets.

A complete-intersection flag on $X$ first supplies sets $S_k$ whose
cardinalities are $h^0(X,kL)$. They lie in the dilates of a simplex
\[
 \Delta(a_1,\ldots,a_n)
 =\left\{x\in\R_{\ge0}^n:\sum_{i=1}^n\frac{x_i}{a_i}\le1\right\}
\]
of volume $V/n!$. Jet surjectivity for the corresponding monomials can be
transferred back to sections on $X$.

The main construction replaces any two intercepts $A,B$ of the bounding
simplex by $w,AB/w$, for any sufficiently small positive real $w$.
It preserves the number of exponents, the multiplication condition, and
the implication from jet surjectivity after the replacement to jet
surjectivity before it. The other exponent coordinates are carried along
unchanged. The planar node bound and primitive-direction argument come
from the surface method; the real-intercept replacement and its compatibility
with these remaining coordinates allow the process to be iterated in
higher dimensions.

For a fixed sufficiently large integer $r$, choose $w=(V/r)^{1/n}$.
Repeated replacements produce sets $F_k$ inside dilates of
\[
 P_* = \Delta(w,\ldots,w,rw),
 \qquad \#F_k=h^0(X,kL),
 \qquad F_k+F_l\subset F_{k+l}.
\]
Because this simplex still has volume $V/n!$, only $o(k^n)$ of its degree-$k$
lattice points are missing. The multiplication condition upgrades this
density statement: for each fixed compact set $K\subset\operatorname{int}P_*$,
every lattice point of $kK$ belongs to $F_k$ for all sufficiently large $k$.
Indeed, such a point has on the order of
$k^n$ decompositions into two nearly equal degrees, whereas the holes can
exclude only $o(k^n)$ of them. This short argument is a full-density version
of the interior approximation results for semigroups
\cite[Theorem~1.6]{KavehKhovanskii2012}.

The long last intercept of $P_*$ permits explicit interpolation at $r$
points that differ only in their last coordinate. One-variable Hermite
interpolation, applied to the coefficients of monomials in the other
coordinates, gives arbitrary jets through degree $\lfloor k\theta\rfloor$
for every fixed $0<\theta<w$ and all sufficiently large $k$.
Transferring these tests to $X$, taking a countable intersection of
nonempty Zariski-open loci, and then letting $\theta$ increase to $w$
proves Theorem~\ref{thm:main}.

Section~\ref{sec:transfers} establishes the monomial operations and their
jet transfers, and Section~\ref{sec:flag} constructs the initial exponent
sets on $X$. Section~\ref{sec:reshaping} proves the two-coordinate
replacement. Section~\ref{sec:saturation} establishes interior filling and
interpolation, and Section~\ref{sec:conclusion} assembles the very-general
conclusion.
==== END Maximal-Multipoint-Seshadri-Constants-in-Higher-Dimensions//sections/introduction.tex ====

==== BEGIN Maximal-Multipoint-Seshadri-Constants-in-Higher-Dimensions//sections/reshaping.tex ====
\section{Reshaping two intercepts}\label{sec:reshaping}

We now change two intercepts of a simplex while preserving its volume, the
cardinality of every exponent set, and the graded inclusion.  The replacement
will also preserve the implication from monomial jet surjectivity back to the
original spaces.  The important freedom is that the new small intercept can
be prescribed as a real number; the argument can therefore be iterated even
when the preceding step has produced irrational intercepts.

The node, diagonal compression, and primitive-direction choice adapt the
corresponding constructions of
\citet[Propositions~3.2 and~3.4 and Lemma~4.3]{OpenAI2026Surfaces}.
Here they are applied directly to polynomial spaces in two variables, with
the remaining exponents retained as labels.  We give the argument in full.

\begin{proposition}[Two-intercept reshaping]\label{prop:reshape}
Let $n\ge2$, let $A,B,a_3,\ldots,a_n$ be positive real numbers, and let
$(S_k)_{k\ge1}$ be finite exponent sets satisfying
\[
 S_k\subset k\Delta(A,B,a_3,\ldots,a_n)\cap\Z^n,
 \qquad S_k+S_l\subset S_{k+l}\quad(k,l\ge1).
\]
Set $H=AB$ and $d=\max(3/A,2/B)$.  For every real number $w>0$ with
$dw<1/16$, there are finite sets $(S'_k)_{k\ge1}$ such that
\begin{align}
 S'_k&\subset k\Delta(w,H/w,a_3,\ldots,a_n)\cap\Z^n,
                                                        \label{eq:reshape-bound}\\
 \#S'_k&=\#S_k,
 \qquad S'_k+S'_l\subset S'_{k+l}.                       \label{eq:reshape-graded}
\end{align}
Moreover, for every $k\ge1$, $m\ge0$, and $r\ge1$, if $\M(S'_k)$ surjects
onto jets through degree $m$ at some $r$ distinct points of $\Torus^n$,
then $\M(S_k)$ does so at some $r$ distinct points of $\Torus^n$.
\end{proposition}

\input{figures/reshaping}

\begin{proof}
Fix the target $w$.  We first use a node to replace the triangular bound
$\Delta(A,B)$ by a quadrilateral of the same area.  A diagonal compression
then produces a triangle with a variable shape.  We choose that shape and a
primitive lattice direction together so that two further compressions give
exactly $\Delta(w,H/w)$.

Throughout, write $\gamma=(\gamma_3,\ldots,\gamma_n)$ for an exponent in
the unchanged coordinates.  For a label occurring in degree $k$, the pair
exponents with this label lie in
\begin{equation}\label{eq:residual-degree}
 \kappa\Delta(A,B),\qquad
 \kappa=k-\sum_{i=3}^n\frac{\gamma_i}{a_i}\ge0.
\end{equation}
For $n=2$ the label is empty and $\kappa=k$.  Each operation below acts on
these two-coordinate spaces, carrying the common monomial with exponent
$\gamma$ along unchanged.  In particular, a planar bound by $\kappa K$
will hold simultaneously for every label.  If $\kappa=0$, the only possible
pair exponent is zero and it stays zero throughout; hence the geometric
calculations may be made with $\kappa>0$.

\paragraph{The node and its exponent bound.}
In pair variables $x,y$, consider
\begin{equation}\label{eq:node-polynomial}
 g(x,y)=y^2-x^2(1+x).
\end{equation}
This polynomial is irreducible, since $1+x$ is not a square in $\C(x)$.
Its curve has the parametrization
\begin{equation}\label{eq:node-parametrization}
 x=v^2-1,\qquad y=v(v^2-1),
\end{equation}
which covers the part where $x\ne0$.  Near the origin choose the analytic
square root with value $1$ and set
\begin{equation}\label{eq:node-coordinates}
 \xi=y+x\sqrt{1+x},\qquad
 \eta=y-x\sqrt{1+x}.
\end{equation}
These are local coordinates and $g=\xi\eta$.

For now fix a real parameter $t>0$, and order the Taylor exponents $(\alpha,\beta)$
in $\xi,\eta$ by lexicographic comparison of
$(\alpha+t\beta,\beta)$, and denote the least exponent by $\nu_t$.
This is a multiplicative well-order: only finitely many nonnegative integer
exponents have weight below any given bound.  The node gives two controls
on these exponents: its two branches bound the least weight after factors
of $g$ have been removed, while each extracted factor translates the least
exponent by $(1,1)$.

Let $f\ne0$ be supported in $\kappa\Delta(A,B)$, and write $f=g^j f_0$,
where $j\ge0$ and $g$ does not divide $f_0$.
The weighted polynomial degree with weights $(1/A,1/B)$ is additive on
nonzero products, since highest-weight forms multiply nontrivially.  As
$g$ has weighted degree $d$, the weighted degree of $f_0$ is at most
$\kappa-jd$, which is nonnegative.  Under
\eqref{eq:node-parametrization}, its restriction is a nonzero polynomial
in $v$: otherwise $f_0$ would vanish on the irreducible curve $g=0$.
Define
\begin{equation}\label{eq:normalization-degree-constant}
 W=\max(2A,3B)=dH.
\end{equation}
The monomial inequality
\[
 2i+3i'\le W\left(\frac{i}{A}+\frac{i'}{B}\right)
\]
shows that the restriction polynomial has degree at most
\begin{equation}\label{eq:restriction-degree}
 (\kappa-jd)W.
\end{equation}

Let $\mu$ be the weight of $\nu_t(f_0)$.  At $v=1$, the branch is
$\eta=0$ and $\xi$ vanishes simply.  At $v=-1$, the branch is $\xi=0$
and $\eta$ vanishes simply.  Indeed, the chosen square root becomes $v$
near $1$ and $-v$ near $-1$, and the nonzero branch coordinate in either
case is $2v(v^2-1)$.  Every nonzero pure $\xi$ term in $f_0$ has exponent
at least $\mu$, and every nonzero pure $\eta$ term has exponent at least
$\mu/t$.  The restriction polynomial therefore has orders at least
$\mu$ and $\mu/t$ at these two distinct points.  Comparing their sum with
\eqref{eq:restriction-degree} gives
\begin{equation}\label{eq:two-branch-bound}
 \left(1+\frac1t\right)\mu\le(\kappa-jd)W.
\end{equation}
To express this bound as a triangle, define its intercepts
\begin{equation}\label{eq:node-intercepts}
 a=\frac{Wt}{1+t},\qquad b=\frac{W}{1+t}.
\end{equation}
For $\nu_t(f_0)=(\alpha,\beta)$, inequality~\eqref{eq:two-branch-bound}
is exactly $\alpha/a+\beta/b\le\kappa-jd$.  Hence
$\nu_t(f_0)\in(\kappa-jd)\Delta(a,b)$.

The extracted factors account for the rest of the degree bound.  Set
$c=1/d$, so their contribution is
$\nu_t(g^j)=(j,j)=jd(c,c)$.  The coefficients $\kappa-jd$ and $jd$
sum to $\kappa$, and therefore
\begin{equation}\label{eq:node-envelope}
 \begin{split}
 \nu_t(f)&\in(\kappa-jd)\Delta(a,b)+jd\{(c,c)\}\subset\kappa Q,\\
 Q&=\conv\{(0,0),(a,0),(c,c),(0,b)\}.
 \end{split}
\end{equation}
Thus the axis triangle comes from the two branch orders, and the diagonal
point comes from powers of the node equation.

From this point on, restrict the free parameter to $t>T$, where
\begin{equation}\label{eq:node-parameters}
 T=d^2H=\max(9B/A,4A/B)\ge6.
\end{equation}
The inequality
\[
 \frac ca+\frac cb=\frac{(1+t)^2}{Tt}>1
\]
shows that $Q$ is a quadrilateral with the vertices in the stated order.
Its area is
\begin{equation}\label{eq:node-area}
 \operatorname{area}(Q)=\frac{c(a+b)}2=\frac H2.
\end{equation}
The identity $W=dH$ is what makes this area equal to that of the input
triangle $\Delta(A,B)$.  The parameter $t$ remains free to change the
shape of $Q$ for the later lattice-direction choice.
For each label, replace its pair support by the possible values of
$\nu_t$ on its polynomial span.  The product of the spaces in degrees
$k,l$ with labels $\gamma,\gamma'$ lies in the space of degree $k+l$
with label $\gamma+\gamma'$, by the original support inclusion.
Lemmas~\ref{lem:least-terms}
and~\ref{lem:row-valuation} preserve cardinality and the graded inclusion,
and \eqref{eq:node-envelope} supplies the bound $\kappa Q$.

We also need the jet implication for this change.  At each fixed degree,
choose a basis with distinct least exponents in each nonempty label space.
One sufficiently small $e>0$ makes weights $(1,t+e)$ select all these
least terms strictly.  To see this, choose $M$ larger than all the selected
$(1,t)$-weights.  Among the finitely many exponents of weight at most $M$,
a small positive perturbation preserves strict comparisons and realizes
the tie-break by the second exponent.  Make $e$ smaller still so that all
selected weights remain below $M$; exponents originally above $M$ cannot
interfere, because a positive perturbation only increases their weights.
Lemmas~\ref{lem:weighted-limit}
and~\ref{lem:rank-transfer} apply under
$(\xi,\eta)=(sX,s^{t+e}Y)$, with the other variables unchanged.
Their inverse images are torus points.  In fact, for each of the finitely
many test points with $X,Y\ne0$, one has $\eta/\xi\to0$, while
\[
 x=\frac{\xi-\eta}{2}+O\bigl((\xi-\eta)^2\bigr),
 \qquad y=\frac{\xi+\eta}{2}.
\]
Thus $x,y\ne0$ for all sufficiently small positive $s$.  The local
coordinate map is injective on a common neighborhood, so it also preserves
distinctness of the test points.

\paragraph{From the quadrilateral to a triangle.}
Apply the integral shear
\[
 (\alpha,\beta)\longmapsto(\alpha-\beta,\beta)
\]
and then compress in the second coordinate.  The transformed quadrilateral
has vertices $(0,0),(a,0),(0,c),(-b,b)$.  For first coordinate $q$ in
$[-b,0]$, its lower boundary is $-q$ and its upper boundary is
$c+(c-b)q/b$; their difference is $c(1+q/b)$.  For $q\in[0,a]$, the
vertical slice length is $c(1-q/a)$.  Compression therefore gives the
triangle
\begin{equation}\label{eq:compressed-triangle}
 P=\conv\{(-b,0),(a,0),(0,c)\},\qquad
 \operatorname{area}(P)=\frac H2.
\end{equation}
Figure~\ref{fig:node-compression} displays these slice lengths geometrically.
Lemmas~\ref{lem:unimodular} and~\ref{lem:compression} show that the exponent
sets remain graded, retain their cardinalities, admit backward jet
transfer, and are bounded, label by label, by $\kappa P$.

\ReshapingNodeFigure

The area is now correct, but we have not yet obtained the prescribed
intercept $w$.  We choose a primitive integral direction whose transverse
range is exactly $H/w$.  In the resulting coordinates, the longest
horizontal slice will have length $w$; compression will turn that length
into the horizontal intercept.

\paragraph{Choosing the primitive direction.}
Put $R=H/w$ and $a_0=WT/(1+T)$.  As $t$ ranges over $(T,\infty)$,
the intercept $a$ ranges over $(a_0,W)$.  Choose $j$ to be the largest
power of two not exceeding
\[
 \frac{R}{2W}=\frac{1}{2dw}.
\]
Then
\begin{equation}\label{eq:primitive-j}
 j>\frac{1}{4dw}>4,
 \qquad R-jW\ge\frac R2>0.
\end{equation}
The open interval
\begin{equation}\label{eq:primitive-interval}
 \bigl(d(R-jW),\ d(R-ja_0)\bigr)
\end{equation}
has length
\[
 \frac{djW}{1+T}=\frac{jT}{1+T}>2.
\]
It therefore contains an odd integer $i$.  This is the point of choosing
$j$ to be a power of two: the vector $u=(i,j)$ is automatically primitive.
Choose the unique $t>T$ for which
\begin{equation}\label{eq:tuned-intercept}
 a=\frac{R-i/d}{j}.
\end{equation}
The interval \eqref{eq:primitive-interval} ensures that this value lies in
$(a_0,W)$.

At the three vertices $(-b,0),(a,0),(0,c)$ of $P$, the functional
$z\mapsto\det(u,z)$ takes values $jb,-ja,i/d$, respectively.  They are
distinct and ordered as
\begin{equation}\label{eq:vertex-heights}
 -ja<jb<i/d,
 \qquad i/d-jb=R-jW>0.
\end{equation}
Their range is exactly $i/d+ja=R$.  This choice used only inequalities
between real numbers; in particular, no rationality of the intercepts was
required.  We now fix this $t$ and this primitive vector for all degrees.

\paragraph{The two final compressions.}
Complete $u$ to an integral basis $(u,u')$ with determinant $1$, and use
the coefficients in that basis as new exponent coordinates.  The second
coordinate of a vector $z$ is $\det(u,z)$.  Thus the transformed triangle
still has area $H/2$, and its three vertex heights, say
$q_0<q_1<q_2$, satisfy $q_2-q_0=R$.

Its horizontal slice length is zero at $q_0$, increases linearly up to
$q_1$, and decreases linearly to zero at $q_2$.  If the maximum slice
length is $h_{\max}$, integration of this tent gives area
$Rh_{\max}/2$.  Hence $h_{\max}=H/R=w$.  Compressing in the first
coordinate therefore gives
\begin{equation}\label{eq:tent-triangle}
 \conv\{(0,q_0),(w,q_1),(0,q_2)\}.
\end{equation}
Its vertical slice at first coordinate $h\in[0,w]$ has length
$R(1-h/w)$.  A final compression in the second coordinate produces
$\Delta(w,R)$, as illustrated in Figure~\ref{fig:final-compressions}.

\ReshapingCompressionFigure

All these changes satisfy Lemmas~\ref{lem:unimodular}
and~\ref{lem:compression}.  They commute with scaling of the bounding
shapes, so for the label $\gamma$ the final pair exponents lie in
$\kappa\Delta(w,R)$.  Substituting \eqref{eq:residual-degree} gives
\eqref{eq:reshape-bound}.  Cardinality, the graded inclusion, and the
backward jet implication hold at every stage, proving all the assertions.

For clarity, the sets in every degree are defined by the fixed exact order
$\nu_t$, the fixed integral maps, and exact orders of vanishing at $1$.
The perturbation $e$ and the small scaling parameters only witness
jet-rank implications at a specified degree.  They may depend on that
degree and on the test tuple without changing the sets or their graded
inclusion.
\end{proof}
==== END Maximal-Multipoint-Seshadri-Constants-in-Higher-Dimensions//sections/reshaping.tex ====

==== BEGIN Maximal-Multipoint-Seshadri-Constants-in-Higher-Dimensions//transfers.tex ====
\section{Monomial spaces and finite jets}
\label{sec:transfers}

The proof will replace spaces of sections by monomial spaces and then change
their exponent sets. Two properties must survive these changes: multiplication
between different degrees, and the implication that interpolation for the new
space yields interpolation for the old one. We establish these properties
separately. Multiplication concerns the exact exponent sets in every degree;
interpolation will be transferred by analytic limits at one fixed degree.

\subsection{Jets and graded exponent sets}

Write $\Torus^n=(\C^*)^n$. For a finite set $S\subset\Z^n$, put
\[
 \M(S)=\Span_{\C}\{x^\alpha:\alpha\in S\},
 \qquad x^\alpha=x_1^{\alpha_1}\cdots x_n^{\alpha_n}.
\]
These Laurent polynomials are regular on $\Torus^n$. If $A$ is a line bundle
on a smooth complex variety $Y$, its jets through an integer degree $m\geq0$ at $p\in Y$
form the vector space
\[
 \Jet_p^m(A)
 =A_p/\frakm_p^{m+1}A_p.
\]
Here $A_p$ denotes the stalk of the sheaf of sections. A space of sections
$W$ \emph{separates jets through degree $m$} at distinct points
$p_1,\ldots,p_r$ if the evaluation map
\[
 W\longrightarrow\bigoplus_{i=1}^r\Jet_{p_i}^m(A)
\]
is surjective. For functions we use the trivial line bundle. Local frames and
local coordinates change the matrices of these maps but not their ranks.
Analytic and algebraic finite jets agree at a smooth complex point, since
the corresponding local rings have the same completion.
Our aim is to obtain such surjections for $W=H^0(X,kL)$ with $m/k$
approaching $(V/r)^{1/n}$.

\begin{definition}\label{def:graded-support}
A \emph{graded support family} in $\Z^n$ is a sequence of finite sets
$(S_k)_{k\geq1}$ such that
\begin{equation}\label{eq:graded-support}
 S_k+S_l\subset S_{k+l}\qquad(k,l\geq1).
\end{equation}
Empty sets are allowed. Equivalently,
$\M(S_k)\M(S_l)\subset\M(S_{k+l})$, where a product of spaces denotes the
span of their pairwise products.
\end{definition}

We will say that an operation on monomial spaces \emph{transfers jet
separation backward} if, for every fixed degree, number of points, and jet
order, separation by the output space at some distinct torus tuple implies
separation by the input space at some distinct torus tuple. The tuples need
not be the same. A finite sequence of such operations has the same property.

\subsection{Least terms and analytic limits}

A monomial order on $\Z_{\geq0}^q$ will mean a total well-order preserved by
addition. For a nonzero convergent power series $f$, let $\nu(f)$ be the
least exponent in its support for this order.

\begin{lemma}[Least-term bases]\label{lem:least-terms}
Let $W$ be a finite-dimensional space of convergent power series in $q$
variables, and fix a monomial order. The set
\[
 \nu(W\setminus\{0\})
\]
has cardinality $\dim W$. There is a basis of $W$ whose least exponents are
exactly this set, with each least coefficient equal to $1$. Moreover,
$\nu(fg)=\nu(f)+\nu(g)$ for nonzero convergent power series $f,g$.
\end{lemma}

\begin{proof}
If $W\ne0$, choose the smallest exponent occurring as the least exponent of
an element of $W$, and normalize one such element to have coefficient $1$
there. Continue in the kernel of that coefficient functional, which has
codimension one. After $\dim W$ steps this gives a basis with strictly
increasing least exponents. In any nonzero linear combination, the first
basis element with a nonzero coefficient determines the least term. This
proves the assertion about the whole set of possible exponents.

For the product assertion, write $a=\nu(f)$ and $b=\nu(g)$. Every exponent
of $f$ is at least $a$, and every exponent of $g$ is at least $b$. Additivity
of the order implies that any pair other than $(a,b)$ has sum strictly
greater than $a+b$. The coefficient of $x^{a+b}$ in the product is therefore
the product of the two nonzero least coefficients.
\end{proof}

\begin{lemma}[Least terms in labeled spaces]\label{lem:row-valuation}
Fix integers $1\leq q\leq n$ and a monomial order in $q$ variables. For
$k\geq1$ and $\gamma\in\Z^{n-q}$, let $W_{k,\gamma}$ be
finite-dimensional spaces of convergent power series in those variables,
with only finitely many nonzero spaces in each degree. Assume that
\[
 W_{k,\gamma}W_{l,\eta}\subset W_{k+l,\gamma+\eta}
 \qquad(k,l\geq1;\ \gamma,\eta\in\Z^{n-q}).
\]
Then
\[
 S'_k=\{(\nu(f),\gamma):0\ne f\in W_{k,\gamma}\}
\]
is a graded support family and
$\#S'_k=\sum_\gamma\dim W_{k,\gamma}$.
\end{lemma}

\begin{proof}
The cardinality assertion follows from Lemma~\ref{lem:least-terms} in each
label. Given two exponents in $S'_k$ and $S'_l$, choose functions witnessing
them. Their nonzero product belongs to the summed degree and label, and
its least exponent is the sum of the two least exponents.
Expanding products of arbitrary linear combinations therefore gives
$\M(S'_k)\M(S'_l)\subset\M(S'_{k+l})$. The full new monomial spaces
thus retain the multiplication property needed for the next operation;
no simultaneous choice of lifts to earlier spaces is required.
\end{proof}

The finite-degree passage to initial monomials is a standard
jet-degeneration principle; see \citet[Proposition~5.7]{Ito2013}. The next
two lemmas give the direct analytic form used here, including unchanged
coordinate factors. The first is analytic; the second is finite-dimensional
linear algebra.

\begin{lemma}[Weighted limits with unchanged variables]
\label{lem:weighted-limit}
Let $1\leq q\leq n$, and let $f_1,\ldots,f_N$ be convergent power series
near $0\in\C^q$. Suppose
that a vector $\lambda\in\R_{>0}^q$ makes a specified exponent $e_j$ the
unique minimum of $\lambda\cdot\alpha$ on the support of $f_j$, and let
$c_j\ne0$ be its coefficient. For any
$\gamma_j\in\Z^{n-q}$, as $s\downarrow0$ the functions
\begin{equation}\label{eq:weighted-limit}
 c_j^{-1}s^{-\lambda\cdot e_j}
 f_j(s^{\lambda_1}Z_1,\ldots,s^{\lambda_q}Z_q)U^{\gamma_j}
 \longrightarrow Z^{e_j}U^{\gamma_j}
\end{equation}
converge locally uniformly on $\C^q\times\Torus^{n-q}$, with all derivatives
of any fixed finite order. The assertion means that each fixed compact set
is in the domain of the left-hand side for all sufficiently small $s>0$.
When $q=n$, the unchanged variable block is omitted.
\end{lemma}

\begin{proof}
First omit the unchanged factor $U^{\gamma_j}$. On a bounded polydisc in
$Z$, choose $s_0>0$ small enough that the power series converges absolutely
on a slightly larger polydisc after scaling by $s_0^\lambda$. In the
normalized expansion every exponent $\lambda\cdot(\alpha-e_j)$ is
nonnegative, and every exponent other than the least one is strictly
positive. For $0<s\leq s_0$, the terms are
dominated by the summable series obtained at $s_0$. Dominated convergence
gives locally uniform convergence to $Z^{e_j}$. Applying Cauchy's estimates
on slightly larger polydiscs gives convergence of derivatives. The unchanged
Laurent monomials and their derivatives are holomorphic near every compact
subset of the torus, so multiplying by them preserves these conclusions.
\end{proof}

\begin{lemma}[Persistence of jet separation]\label{lem:rank-transfer}
Fix a finite tuple of distinct points in a complex manifold and a jet order
$m$. Suppose that finitely many holomorphic functions, depending on a
positive parameter $s$, converge with their derivatives through order $m$
near that tuple. If the limiting functions separate those jets, then the
functions for sufficiently small $s>0$ do also.

The same conclusion applies to sections written in local frames. If the
functions before the limit are pullbacks under a coordinate change sending
the tuple to distinct points, their original functions separate jets at the
image tuple. Multiplication of individual functions by nonzero constants
does not affect this statement.
\end{lemma}

\begin{proof}
Choose coordinates and frames near the points. The entries of the jet matrix
are the finitely many required Taylor coefficients. A maximal minor that is
nonzero in the limit remains nonzero for small $s$. Coordinate changes and
frame changes induce isomorphisms of truncated local rings or their
rank-one modules. Rescaling an individual function rescales a matrix column
by a nonzero constant.
\end{proof}

In particular, the unchanged variables in
Lemma~\ref{lem:weighted-limit} cause no restriction on the jets being
separated: all derivatives, including derivatives in those variables, enter
the same convergent jet matrix. We shall apply the lemmas to finitely many
spaces distinguished by a common exponent in the unchanged variables.

\subsection{Lattice changes and coordinate compression}

We next record two operations directly on Laurent monomials. They will let
us change the shape of an exponent set while retaining both properties
above.

\begin{lemma}[Unimodular changes]\label{lem:unimodular}
Let $B\in\operatorname{GL}_n(\Z)$. Replacing $S_k$ by $BS_k$ preserves
cardinality and the graded support property. It preserves, in both
directions, the existence of a distinct torus tuple at which the associated
monomial space separates any prescribed finite jets.
\end{lemma}

\begin{proof}
The lattice map is bijective and additive. The torus automorphism
\[
 \phi_B(x)_j=\prod_{i=1}^n x_i^{B_{ij}}
\]
has pullback $\phi_B^*(x^\alpha)=x^{B\alpha}$. Its inverse is the
monomial map associated with $B^{-1}\in\operatorname{GL}_n(\Z)$, so the
jet assertion follows by change of coordinates.
\end{proof}

For compression in the first coordinate, write an exponent as $(a,\beta)$
with $a\in\Z$ and $\beta\in\Z^{n-1}$. For a finite set $S$, define its
row space at $\beta$ by
\[
 W_\beta(S)=\Span_\C\{z^a:(a,\beta)\in S\},
\]
and define the compressed exponent set by
\begin{equation}\label{eq:coordinate-compression}
 \mathcal C_1(S)=
 \{(\ord_{z=1}f,\beta):0\ne f\in W_\beta(S)\}.
\end{equation}
All functions in a row are holomorphic near $z=1$, even when their
exponents are negative. Compression in another coordinate is defined by
permuting coordinates.

\begin{lemma}[Coordinate compression]\label{lem:compression}
Coordinate compression preserves the cardinalities and the graded support
property of a family $(S_k)$. It transfers jet separation backward. In a
nonempty row with smallest and largest exponents $a_-$ and $a_+$, every
compressed exponent lies in $[0,a_+-a_-]$.

Consequently, suppose that, after fixing all but two coordinates, the pair
exponents lie in a scaled convex polygon $\kappa Q$, where $\kappa\geq0$.
Compression in one pair coordinate bounds the new pair exponents by the
shape obtained by moving each interval slice parallel to that coordinate
to start at zero, without changing its length. This shape operation commutes
with scaling by $\kappa$.
\end{lemma}

\begin{proof}
Use the coordinate $u=z-1$ near $z=1$. Lemma~\ref{lem:least-terms}
identifies the possible orders in a row with a basis of distinct leading
orders. If the input sets are graded, the product of two row spaces lies in
the summed degree and remaining exponent. Lemma~\ref{lem:row-valuation}
therefore proves both cardinality preservation and the graded support
property.

Multiplying a nonzero row element by the unit $z^{-a_-}$ gives a polynomial
of degree at most $a_+-a_-$. Its order at $1$ is unchanged and cannot exceed
that degree. This proves the row bound, including for Laurent exponents.

For a fixed degree, choose in every nonempty row a basis with distinct
orders and leading coefficients $1$ in $u$. Substitute $z=1+sZ$, normalize
each basis element by $s$ to its order, and carry along the unchanged
monomial in the other variables. Lemmas~\ref{lem:weighted-limit}
and~\ref{lem:rank-transfer} apply to all rows simultaneously. At a fixed
finite torus tuple, the substitution has nonzero values and is injective
for sufficiently small $s>0$. It gives a local coordinate change, so jet
separation transfers to the original space at the image tuple.

Finally, a row contained in an interval slice of $\kappa Q$ has exponent
span no greater than the length of that slice. Its compressed row is
therefore contained in the interval beginning at zero with that length.
For $\kappa>0$, slices and their lengths scale by $\kappa$. When
$\kappa=0$, the only possible pair exponent is zero and compression leaves
it unchanged.
\end{proof}

When these operations act on a graded family, the support sets are defined
by fixed exact orders and fixed lattice maps. The weights and small
parameters used to transfer jets may nevertheless depend on the degree:
they only establish finite-rank implications. The multiplication property
in Lemma~\ref{lem:row-valuation} holds independently of those analytic
choices.
==== END Maximal-Multipoint-Seshadri-Constants-in-Higher-Dimensions//transfers.tex ====

==== BEGIN Maximal-Multipoint-Seshadri-Constants-in-Higher-Dimensions//triangle.tex ====
\begin{figure}[htbp]
\centering
\begin{tikzpicture}[x=1.18cm,y=.92cm,font=\small,
  line cap=round,line join=round,>=stealth]
  \definecolor{sliceblue}{RGB}{40,88,125}
  \definecolor{sliceorange}{RGB}{161,80,28}
  \fill[sliceblue!9] (.44,1.2)--(1.1,3)--(2.24,1.2)--cycle;
  \draw[->,gray!70] (-.35,0)--(3.65,0) node[right,text=black] {$y$};
  \draw[->,gray!70] (0,-.12)--(0,3.6) node[above,text=black] {$\ell(y)$};
  \draw[sliceblue,very thick] (0,0)--(1.1,3)--(3,0);
  \draw[densely dashed,gray!65] (0,3)--(1.1,3);
  \draw[sliceorange,thick] (0,1.2)--(3.28,1.2);
  \draw[densely dashed,gray!65] (.44,0)--(.44,1.2)
    (2.24,0)--(2.24,1.2) (1.1,0)--(1.1,3);
  \fill[sliceorange] (.44,1.2) circle (1.7pt) (2.24,1.2) circle (1.7pt);
  \node[left,inner sep=4pt] at (0,3) {$\rho$};
  \node[left,inner sep=4pt] at (0,1.2) {$e$};
  \node[below,inner sep=4pt] at (0,0) {$y_0$};
  \node[below,inner sep=4pt] at (1.1,0) {$y_1$};
  \node[below,inner sep=4pt] at (3,0) {$y_2$};
  \draw[<->,sliceorange] (.44,-.68)--(2.24,-.68)
    node[midway,fill=white,inner sep=3pt] {$\rho-e$};
  \draw[gray!55] (.44,-.45)--(.44,-.83) (2.24,-.45)--(2.24,-.83);
  \draw[<->] (0,-1.20)--(3,-1.20)
    node[midway,fill=white,inner sep=3pt] {$\rho$};
  \draw[gray!55] (0,-.93)--(0,-1.35) (3,-.93)--(3,-1.35);
\end{tikzpicture}
\caption{Horizontal slice lengths of a triangle of area $\rho^2/2$
and vertical range $\rho$. The tent has height $\rho$. For
$0\le e\le\rho$, the heights in the vertical projection at which
$\ell(y)\ge e$ form an interval of length $\rho-e$.
The graph shows three distinct vertex heights; a horizontal side gives
the same conclusion with a one-sided tent.}
\label{fig:triangle-slices}
\end{figure}
==== END Maximal-Multipoint-Seshadri-Constants-in-Higher-Dimensions//triangle.tex ====

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