64c3814
==== TABLE OF CONTENTS ====
Maximal-multipoint-Seshadri-constants-in-positive-characteristic//01-introduction.tex
Maximal-multipoint-Seshadri-constants-in-positive-characteristic//02-transfer.tex
Maximal-multipoint-Seshadri-constants-in-positive-characteristic//03-flag.tex
Maximal-multipoint-Seshadri-constants-in-positive-characteristic//04-reshape.tex
Maximal-multipoint-Seshadri-constants-in-positive-characteristic//05-interpolation.tex
Maximal-multipoint-Seshadri-constants-in-positive-characteristic//06-positivity.tex
Maximal-multipoint-Seshadri-constants-in-positive-characteristic//main.tex
Maximal-multipoint-Seshadri-constants-in-positive-characteristic//preamble.tex
Maximal-multipoint-Seshadri-constants-in-positive-characteristic//references.bib
===========================
==== BEGIN Maximal-multipoint-Seshadri-constants-in-positive-characteristic//01-introduction.tex ====
\section{Introduction}\label{sec:introduction}
Multipoint Seshadri constants measure how much positivity an ample line
bundle retains after imposing simultaneous conditions at several points.
Their volume bound depends only on the top self-intersection of the line
bundle and the number of points. The question is whether, for sufficiently
many general points, any curve can impose a stricter bound. We prove that
no such curve exists at a geometric generic tuple in positive
characteristic.
Let $k$ be an algebraically closed field of characteristic $p>0$, let
$X/k$ be a smooth integral projective variety of dimension $n$, and let
$L$ be an ample line bundle. For $r\ge1$, write
\[
U_r=\{(x_1,\ldots,x_r)\in X^r:x_i\ne x_j\text{ for }i\ne j\}
\]
for the open configuration space of ordered distinct points. Put
$K_r=\overline{k(U_r)}$, and let $p_1,\ldots,p_r\in X(K_r)$ be the
points specified by its geometric generic point. Define
\begin{equation}\label{eq:generic-seshadri}
\varepsilon_r(X,L)=\inf_C
\frac{L_{K_r}\cdot C}{\sum_{i=1}^r\mult_{p_i}C},
\end{equation}
where $C$ ranges over integral curves in $X_{K_r}$ meeting at least one
$p_i$. Multiplicity at a point outside $C$ is zero. Throughout, we use
this ordinary, curve-defined Seshadri constant.
\begin{theorem}\label{thm:main}
Let $X/k$ be a smooth integral projective variety over an algebraically
closed field of characteristic $p>0$, with $\dim X=n\ge3$, and let $L$
be ample. There is an integer $r_0=r_0(X,L)$ such that
\[
\varepsilon_r(X,L)=\left(\frac{L^n}{r}\right)^{1/n}
\qquad\text{for every integer }r\ge r_0.
\]
\end{theorem}
The same conclusion holds when $n=2$; see Remark~\ref{rem:surfaces}.
The geometric generic formulation includes countable algebraically closed
fields. It also gives a precise meaning to general position without
choosing points in an infinite intersection of open subsets. The equality
is exact for each sufficiently large point count. On the blow-up of the
geometric generic points, it says that
\[
\pi^*L_{K_r}-(L^n/r)^{1/n}\sum_{i=1}^r E_i
\]
is a nef real divisor class with top self-intersection zero.
\subsection{History and the characteristic issue}
Nagata's work on Hilbert's fourteenth problem led to his conjecture on
the degrees of plane curves with assigned multiplicities at general
points, and to its proof when the number of points is a square at least $16$
\cite{Nagata1959}. The qualitative Nagata--Biran--Szemberg assertion
extends eventual maximality to polarized varieties. Ro\'e and Ross
formulate it in arbitrary dimension over uncountable algebraically closed
fields and distinguish the ordinary
Seshadri constant from its higher-dimensional cycle analogues
\cite[Conjecture~1.3 and Remark~1.2]{RoeRoss2009}.
Theorem~\ref{thm:main} proves its positive-characteristic form, with
general position expressed by the geometric generic tuple, in the stated
dimensions.
Several preceding results explain the volume scale of this question.
Biran's packing stability theorem gives a symplectic analogue on
closed four-manifolds with rational symplectic class \cite{Biran1999}.
K\"uchle obtains asymptotically optimal lower bounds for multipoint
Seshadri constants on complex projective varieties \cite{Kuechle1996}.
Product inequalities compare
point counts on a variety with those on projective space
\cite[Theorem~1.1 and Remark~1.2]{RoeRoss2009}.
These are distinct from exact algebraic maximality for every sufficiently
large integer $r$.
Our proof adapts the graded-support construction of the companion
\emph{Maximal Multipoint Seshadri Constants in Higher Dimensions}
\cite[Sections~2--6]{OpenAI2026Higher}, which treats complex varieties.
Its two-coordinate reshaping follows the nodal exponent bounds and
primitive lattice directions in the surface companion
\cite[Propositions~3.2 and~3.4 and Lemma~4.3]{OpenAI2026Surfaces}.
We prove every required statement here. The passage to positive
characteristic requires an algebraic replacement for analytic limits and
a nodal equation that retains two distinct branches in characteristic
two. Formal substitutions over $k[[s]]$ supply the first; the curve
$y^2+xy-x^3=0$ supplies the second.
The relation between positivity and asymptotic jet separation is
classical \cite[Theorem~6.4]{Demailly1992}. The use of leading monomials
belongs to the valuation approach to linear series
\cite{Okounkov1996,LazarsfeldMustata2009,KavehKhovanskii2012};
Ito gives finite-degree comparisons between multipoint jet separation
and initial monomial spaces \cite[Proposition~5.7 and Theorem~5.8]{Ito2013}.
Park and Shin give characteristic-independent Okounkov-body descriptions
of one-point Seshadri constants for complete series of nef and big
divisors \cite[Theorem~1.2(1)]{ParkShin2021}.
Monomial interpolation also underlies Dumnicki's diagram-cutting method
\cite[Proposition~13 and Theorem~14]{Dumnicki2007}.
The statements below isolate two useful forms of these ideas: formal
transfer of arbitrary finite jets, and reshaping of a graded family with
prescribed real simplex intercepts. The latter keeps both the exact
cardinality in every degree and multiplication between degrees. A short
counting argument then turns full asymptotic density into exact filling
of compact subsets of the interior.
\subsection{Proof strategy}
Put $V=L^n$ and, for a fixed point count $r$, put $w=(V/r)^{1/n}$.
For a line bundle $A$ at a smooth point $x$, its jets through order $m$
form $A_x/\mathfrak m_x^{m+1}A_x$. Separating those jets at a tuple
means that sections surject onto the direct sum of these spaces. Such
separation by $H^0(X,L^{\otimes N})$ gives the curve inequality
\[
NL\cdot C\ge m\sum_i\mult_{p_i}C.
\]
It therefore suffices to obtain jets through order $\lfloor N\theta\rfloor$
for every fixed $0<\theta<w$ and all sufficiently large $N$.
We encode sections by finite sets $S_N\subset\Z^n$ of leading exponents,
with the multiplication property
\[
S_N+S_M\subseteq S_{N+M}.
\]
Their monomial spaces on $\T^n=(\mathbb G_{m,k})^n$ are
$M(S_N)=\Span_k\{z^\alpha:\alpha\in S_N\}$.
Section~\ref{sec:transfer} shows how jet separation for such monomials
can be transferred to the original functions or sections. The transfer
is a finite-degree rank argument. Its auxiliary weights may depend on
$N$, while the orders defining the graded sets remain fixed.
For positive intercepts $a_1,\ldots,a_n$, write
\[
\Delta(a_1,\ldots,a_n)
=\left\{x\in\R_{\ge0}^n:\sum_i x_i/a_i\le1\right\}.
\]
A complete-intersection flag in Section~\ref{sec:flag} gives
\[
S_N\subseteq N\Delta(1/D,\ldots,1/D,D^{n-1}V),
\qquad \#S_N=h^0(X,L^{\otimes N}),
\]
where $DL$ is very ample. The simplex has volume $V/n!$, equal to the
leading coefficient of the section count.
The main geometric step, Section~\ref{sec:reshape}, replaces two
intercepts $A,B$ by $w,AB/w$ for any sufficiently small prescribed real
$w>0$. It preserves cardinalities, the graded property, and the
implication from jet separation after the replacement to jet separation
before it. For a suitable leading-term order, a node bounds the exponents
by a quadrilateral of the same area. Integral changes of coordinates and successive compressions,
which move interval slices to start at zero, turn this quadrilateral
into the desired triangle. The other coordinates serve only as labels.
For large $r$, repeating this step yields graded supports $F_N$ inside
\[
NP_*,\qquad P_* =\Delta(w,\ldots,w,rw),
\qquad \#F_N=\frac{V}{n!}N^n+o(N^n).
\]
Section~\ref{sec:interpolation} proves that every lattice point in a
fixed compact interior region belongs to $F_N$ in large degree. Indeed,
each such point has on the order of $N^n$ decompositions into two
nearly equal degrees, whereas the missing points exclude only $o(N^n)$
of them. The long final intercept of $P_*$ then permits polynomial
interpolation at $r$ points differing only in their last coordinate.
This interpolation uses the Chinese remainder theorem and is valid in
every characteristic.
Each transferred jet-separation statement holds on a nonempty open
subset of $U_r$, hence at its geometric generic point.
Section~\ref{sec:positivity} converts these statements into the required
lower bound for every curve and proves the matching volume upper bound
on the blow-up.
==== END Maximal-multipoint-Seshadri-constants-in-positive-characteristic//01-introduction.tex ====
==== BEGIN Maximal-multipoint-Seshadri-constants-in-positive-characteristic//02-transfer.tex ====
\section{Leading terms and transfer of jets}\label{sec:transfer}
We first turn spaces of sections into finite sets of exponents. The
essential point is that jet separation by the resulting monomials implies
jet separation by the original sections. We will also make two elementary
changes to the exponent sets while retaining this implication.
These operations follow the support method of
\cite[Section~2]{OpenAI2026Higher}; the formal transfer below makes the
method available in arbitrary characteristic.
Throughout, $k$ is algebraically closed. The arguments of this section
work in every characteristic. For a smooth $d$-dimensional variety $Y$,
a line bundle $A$, and $y\in Y(k)$, the jets through order $m\geq0$ are
\[
J_y^m(A)=A_y/\mathfrak m_y^{m+1}A_y.
\]
A finite-dimensional space $W\subset H^0(Y,A)$ \emph{separates these
jets at} $y_1,\ldots,y_r$ if the evaluation map
$W\to\bigoplus_i J_{y_i}^m(A)$ is surjective. The locus of such
ordered distinct tuples is open in $Y^r$: the jet spaces are the fibers
of the locally free sheaf of principal parts of $A$, of rank
$\binom{d+m}{d}$, and surjectivity is a rank condition. The description
of principal parts by the $m$th infinitesimal neighborhood of the diagonal
also shows compatibility with field extension. Local coordinates for a
smooth variety identify its fibers with truncated polynomial rings.
For the diagonal description and base change, see
\cite[Tags~0H90 and~0H8Z]{StacksProject}. Local freeness follows as well
from the filtration whose graded pieces are
$A\otimes\operatorname{Sym}^j\Omega_Y^1$, $0\leq j\leq m$.
Put $\T^d=(\mathbb G_{m,k})^d$. For a finite set $S\subset\Z^d$, write
\[
M(S)=\Span_k\{x^\alpha:\alpha\in S\}
\subset k[x_1^{\pm1},\ldots,x_d^{\pm1}].
\]
Here and below, auxiliary existence statements about points on a torus
refer to $k$-points.
\begin{definition}\label{def:graded}
A family of finite sets $(S_N)_{N\geq1}$ in $\Z^d$ is \emph{graded} if
\[
S_N+S_{N'}\subset S_{N+N'}\qquad(N,N'\geq1).
\]
A transformation $(S_N)\mapsto(T_N)$ has \emph{backward jet transfer}
if, for every $N,r\geq1$ and $m\geq0$, separation of jets through order
$m$ by $M(T_N)$ at some $r$ distinct torus points implies the same
assertion for $M(S_N)$, possibly at a different tuple.
\end{definition}
\subsection{Least exponents}
A monomial order on $\Z_{\geq0}^h$ means an addition-compatible total
well-order. For a nonzero formal series $f\in k[[z_1,\ldots,z_h]]$,
let $\nu(f)$ be the least exponent occurring in $f$, for a fixed such
order. In particular,
\begin{equation}\label{eq:least-product}
\nu(fg)=\nu(f)+\nu(g).
\end{equation}
Indeed, the product of the two least terms is the unique least term in
the product.
\begin{lemma}\label{lem:least-terms}
For a finite-dimensional subspace $W\subset k[[z_1,\ldots,z_h]]$, the
set $\{\nu(f):0\neq f\in W\}$ has cardinality $\dim_k W$.
More generally, suppose finite-dimensional spaces $W_{N,\gamma}$, with
only finitely many nonzero spaces in each degree, are indexed by
$N\geq1$ and $\gamma\in\Z^{d-h}$ and satisfy
\[
W_{N,\gamma}W_{N',\gamma'}
\subset W_{N+N',\gamma+\gamma'}.
\]
Then recording the pairs $(\nu(f),\gamma)$ for $0\neq f\in W_{N,\gamma}$
gives a graded family, with cardinality $\sum_\gamma\dim_k W_{N,\gamma}$
in degree $N$.
\end{lemma}
\begin{proof}
Choose the smallest exponent occurring as $\nu(f)$ for a nonzero member
of $W$, and choose a member whose coefficient there is $1$. The kernel
of that coefficient functional has codimension one. Repeating in this
kernel gives a basis with distinct least exponents. Every nonzero linear
combination has as its least exponent the smallest least exponent among
the basis vectors with nonzero coefficients. This proves the first
assertion; the second follows label by label and from
\eqref{eq:least-product}.
\end{proof}
For the transfer of jets we need to realize finitely many chosen least
terms by a single positive weight vector. The weights may depend on the
degree; the monomial order defining the graded family will remain fixed.
\begin{lemma}\label{lem:exposing-weights}
Let $f_1,\ldots,f_q$ be nonzero formal series, and let $e_j=\nu(f_j)$.
There is a vector $\lambda\in\Z_{>0}^h$ for which $e_j$ is the unique
exponent of least $\lambda$-weight in $f_j$, simultaneously for all $j$,
in either of the following cases:
\begin{enumerate}[label=\textup{(\roman*)}]
\item the order is lexicographic;
\item $h=2$ and, for a fixed real $t>0$, the order compares
$(\alpha+t\beta,\beta)$ lexicographically.
\end{enumerate}
\end{lemma}
\begin{proof}
In the lexicographic case set $\lambda_h=1$, and successively choose
\[
\lambda_i>\max_j\sum_{a>i}\lambda_a(e_j)_a
\qquad(i=h-1,\ldots,1).
\]
If an exponent is lexicographically greater than $e_j$, its gain at the
first differing coordinate outweighs every possible loss in the later
coordinates.
For (ii), the indicated order is a monomial order because every bounded
weight range contains only finitely many nonnegative integral exponents.
Choose a real bound larger than the weights of all the $e_j$. Among the
finitely many exponents below that bound, replacing $t$ by a sufficiently
close rational $t'>t$ preserves all strict comparisons with the $e_j$
and resolves equal-weight comparisons by increasing $\beta$. Exponents
above the bound cannot interfere: their weights only increase, while the
new weights of the $e_j$ stay below the bound. Clearing the denominator
of $(1,t')$ gives the required positive integral weights.
\end{proof}
\subsection{Formal specialization of jet matrices}
The next lemma supplies the passage from formal leading terms to actual
points. Ito's weighted degeneration argument gives such a jet-rank
comparison over $\mathbb C$ \cite[Proposition~5.7]{Ito2013}. Here
substitution in truncated rings supplies a formal version in arbitrary
characteristic, without differentiation or factorial denominators.
\begin{lemma}[Formal jet transfer]\label{lem:formal-transfer}
Let $1\leq h\leq d$, let $Y$ be a smooth integral $h$-dimensional
$k$-variety, and choose $y\in Y(k)$ with formal coordinates
$z_1,\ldots,z_h$. Let $f_1,\ldots,f_q$ be regular functions on a common
neighborhood of $y$, and let $\gamma_j\in\Z^{d-h}$.
Suppose the expansion of $f_j$ has a term $c_jz^{e_j}$, with
$c_j\neq0$, which is uniquely lowest for one common weight vector
$\lambda\in\Z_{>0}^h$.
If the span of the monomials $Z^{e_j}u^{\gamma_j}$ separates jets
through order $m$ at $r$ distinct points of $\T^h\times\T^{d-h}$,
then the span of $f_ju^{\gamma_j}$ separates the same jets at $r$
distinct points in the product of that neighborhood with $\T^{d-h}$.
The points may be chosen in any prescribed dense open subset of this
product. The statement also holds for sections written in a local
frame of a line bundle.
\end{lemma}
\begin{proof}
Let $(Q_i,U_i)$, $1\leq i\leq r$, be a tuple at which the monomials
separate the stated jets. Shrink the neighborhood of $y$ to an affine
one, denoted $Y^\circ$, on which all the functions are defined. Put
\[
R=k[[s]],\qquad F=k((s)),\qquad
B=R[\delta_1,\ldots,\delta_d]/(\delta_1,\ldots,\delta_d)^{m+1},
\]
and write $\delta=(\delta',\delta'')$ according to the two factors.
For each $i$, substitute
\begin{equation}\label{eq:formal-substitution}
z_a=s^{\lambda_a}(Q_{i,a}+\delta'_a),\qquad
u_b=U_{i,b}+\delta''_b.
\end{equation}
The first substitution is defined on the completed local ring, since
$B$ is $s$-adically complete and every $\lambda_a$ is positive. The
second is defined on Laurent polynomials, since each $U_{i,b}$ is
nonzero. In particular, these substitutions define ring maps from the
coordinate ring of $Y^\circ\times\T^{d-h}$ to $B$.
Use the classes of $\delta^\alpha$, $|\alpha|\leq m$, as an $R$-basis
of $B$. The images of $f_ju^{\gamma_j}$ under all $r$ substitutions
give the $j$th column of a matrix with $r\binom{d+m}{d}$ rows. Divide
this column by $c_js^{\lambda\cdot e_j}$. These normalized columns
still belong to $B^{\oplus r}$, and their reductions modulo $s$ are precisely the jets
of $Z^{e_j}u^{\gamma_j}$ at $(Q_i,U_i)$ in translated coordinates.
The assumed surjectivity supplies a maximal minor nonzero modulo $s$.
The original columns therefore span $B[1/s]^{\oplus r}$ over $F$.
We now interpret this rank statement as jet separation on a variety.
Setting $\delta=0$ in \eqref{eq:formal-substitution} gives actual
$F$-points $P_i$ of $Y^\circ\times\T^{d-h}$. After inverting $s$, the
corresponding ring map factors through the order-$m$ jet algebra of
$(Y^\circ\times\T^{d-h})_F$ at $P_i$: elements nonvanishing at $P_i$
map to units, and its maximal
ideal maps into $(\delta)$. Both this jet algebra and $B[1/s]$ have
dimension $\binom{d+m}{d}$ over $F$. Surjectivity onto the direct sum
of the latter spaces forces each induced map from a jet algebra to be
an isomorphism. It follows that the original functions separate the
order-$m$ jets at the $P_i$ themselves. These points are distinct,
since two repeated centers would make two target blocks factor through
one jet algebra, contradicting surjectivity.
Thus the open jet-separation locus has an $F$-point and is nonempty.
The configuration space is irreducible, so this locus meets the
configuration space of any prescribed dense open subset. A nonempty
open subset of a variety over the algebraically closed field $k$ has a
$k$-point. This proves the assertion. A local frame reduces the
line-bundle version to the one just proved.
\end{proof}
Together, Lemmas~\ref{lem:least-terms}--\ref{lem:formal-transfer} allow
us to replace finite-dimensional spaces by their least exponents, using
either of the two orders above, while preserving dimension and
transferring jet separation back to the original spaces. Labels record
the exponents of variables that are left unchanged.
\subsection{Two elementary transformations}
\begin{lemma}\label{lem:unimodular}
For $A\in\operatorname{GL}_d(\Z)$, the transformation
$S_N\mapsto A S_N$ preserves cardinalities and gradedness, and has
backward jet transfer.
\end{lemma}
\begin{proof}
An invertible integral linear map preserves cardinalities and addition.
Its action on characters is induced by a torus automorphism, which
identifies the corresponding jet evaluations.
\end{proof}
The other transformation uses orders of vanishing to place each row in an
interval starting at zero. The resulting exponents need not be consecutive.
\begin{lemma}[Coordinate compression]\label{lem:compression}
Let $(S_N)$ be a graded family in $\Z^d$, and select one coordinate,
written $z$. For each degree $N$ and each label $\gamma\in\Z^{d-1}$,
form its row space
\[
W_{N,\gamma}
=\Span_k\{z^a:(a,\gamma)\in S_N\}.
\]
Replace this row by the set of orders at $z=1$ of nonzero elements of
$W_{N,\gamma}$, retaining the label $\gamma$. The resulting family
preserves cardinalities and gradedness, and has backward jet transfer.
If the original row lies in a real interval of length $b$, every new
exponent lies in $[0,b]$.
\end{lemma}
\begin{proof}
Expand the row spaces in the parameter $z-1$. Lemma~\ref{lem:least-terms}
proves the cardinality and gradedness assertions, because multiplication
adds both degrees and labels. Lemma~\ref{lem:formal-transfer}, with
$Y=\mathbb G_m$, $y=1$, and weight $1$, proves backward jet transfer.
For a nonempty row let $a_-$ and $a_+$ be its smallest and largest
exponents. Multiplication by $z^{-a_-}$ preserves orders at $1$ and
turns each member into a polynomial of degree at most $a_+-a_-\leq b$.
Its order of vanishing is at most its degree, proving the bound.
\end{proof}
Consequently, suppose a family is bounded by dilates of a polygon.
Replacing each slice of that polygon parallel to the compressed coordinate
by an interval of the same length starting at zero gives a new region
whose dilates contain the compressed family. This applies equally to real
slice endpoints and to every dilation of the polygon. In characteristic
$p$, a compressed row need not contain all intermediate integers: for
example, $\Span_k\{1,z^p\}$ has orders $0$ and $p$ at $1$. Only the
interval bound of Lemma~\ref{lem:compression} will be used.
==== END Maximal-multipoint-Seshadri-constants-in-positive-characteristic//02-transfer.tex ====
==== BEGIN Maximal-multipoint-Seshadri-constants-in-positive-characteristic//03-flag.tex ====
\section{An initial simplex from a flag}\label{sec:flag}
We now apply the leading-term construction to the section ring of $L$.
A complete-intersection flag gives a particularly simple bound on the
exponents: a simplex whose volume equals the leading coefficient of the
Hilbert polynomial. We give the complete argument for the flag
construction of \cite[Section~3]{OpenAI2026Higher}.
For positive real numbers $a_1,\ldots,a_d$, write
\[
\Delta(a_1,\ldots,a_d)
=\left\{x\in\R_{\geq0}^d:
\sum_{i=1}^d\frac{x_i}{a_i}\leq1\right\}.
\]
Its volume is $\prod_i a_i/d!$. We use additive notation $NL$ for
$L^{\otimes N}$.
\begin{proposition}\label{prop:initial-simplex}
Let $X$ be a smooth integral projective $n$-dimensional variety over an
algebraically closed field, with $n\geq2$, and let $L$ be ample. Put
$V=L^n$, and choose an integer $D\geq1$ such that $DL$ is very ample.
There is a graded family of finite sets $S_N\subset\Z_{\geq0}^n$ such
that
\begin{align}
S_N&\subset NP_0,
&P_0&=\Delta(1/D,\ldots,1/D,D^{n-1}V),
\label{eq:initial-simplex}\\
\#S_N&=h^0(X,NL)
=\frac{V}{n!}N^n+o(N^n).
\label{eq:initial-dimension}
\end{align}
For every $N,r\geq1$ and $m\geq0$, if $M(S_N)$ separates jets through
order $m$ at some $r$ distinct torus points, then $H^0(X,NL)$ separates
the same jets at some $r$ distinct points of $X$.
\end{proposition}
\begin{proof}
Bertini's theorem for hyperplane sections, valid in every characteristic
for a projective embedding over an algebraically closed field, gives
sections $\sigma_1,\ldots,\sigma_{n-1}\in H^0(X,DL)$ whose successive
zero loci form a smooth integral flag
\[
X=X_0\supset X_1\supset\cdots\supset X_{n-1}=C_0.
\]
See \cite[Tags~0FD6 and~0G4G]{StacksProject} for the smoothness and
integrality assertions.
At each step the variety being cut has dimension at least two. Choose
$x\in C_0(k)$ and a local frame $e$ of $L$ near $x$. The functions
$z_i=\sigma_i/e^D$, $i<n$, form part of a regular system of parameters
at $x$; complete them by $z_n$. These parameters identify
$\widehat{\mathcal O}_{X,x}$ with $k[[z_1,\ldots,z_n]]$.
Expand every section of $NL$ using the frame $e^N$, and let $S_N$ be
the set of lexicographically least exponents of the nonzero expansions.
The expansion map is injective: a section with zero germ is zero by
integrality, and the local ring injects into its completion.
Lemma~\ref{lem:least-terms} therefore gives
$\#S_N=h^0(X,NL)$ and the graded property. The Hilbert polynomial of
the ample line bundle gives the last equality in
\eqref{eq:initial-dimension}. A basis with distinct least exponents,
Lemma~\ref{lem:exposing-weights}(i), and
Lemma~\ref{lem:formal-transfer} with $h=n$ prove the jet-transfer
assertion.
It remains to prove the simplex bound. Let
$\alpha=(\alpha_1,\ldots,\alpha_n)\in S_N$, coming from a nonzero
section $s$. We successively divide by the equations of the flag and
restrict to the next member. The details matter because the
lexicographic order first measures vanishing along an entire divisor,
not just at the chosen point.
Suppose we have reached a nonzero section on $X_{i-1}$ with least
exponent $(\alpha_i,\ldots,\alpha_n)$ in its formal expansion at $x$.
If $\alpha_i>0$, restriction of this section to $X_i$ has zero formal
expansion. Injectivity into the completed local ring, followed by
integrality of $X_i$, shows that this restriction is identically zero.
The exact sequence for the effective Cartier divisor
$X_i\subset X_{i-1}$ therefore allows division by
$\sigma_i|_{X_{i-1}}$. Repeating this argument $\alpha_i$ times, and
then restricting to $X_i$, gives a nonzero section whose least exponent
is $(\alpha_{i+1},\ldots,\alpha_n)$. Nonvanishing follows from the
nonzero coefficient of the original least term. Each division
subtracts $DL$ from the line bundle.
After these steps, the resulting nonzero section on $C_0$ belongs to
\[
\left(N-D\sum_{i<n}\alpha_i\right)L|_{C_0}
\]
and has order exactly $\alpha_n$ at $x$. Since
$L\cdot C_0=D^{n-1}V$, comparison with the degree of its zero divisor
gives
\[
\alpha_n\leq
\left(N-D\sum_{i<n}\alpha_i\right)D^{n-1}V.
\]
Equivalently,
$D\sum_{i<n}\alpha_i+\alpha_n/(D^{n-1}V)\leq N$, which is
\eqref{eq:initial-simplex}.
\end{proof}
The intercepts of $P_0$ have product $V$. The next section changes
them, two at a time, without changing this product, the numbers of
exponents, or backward jet transfer.
==== END Maximal-multipoint-Seshadri-constants-in-positive-characteristic//03-flag.tex ====
==== BEGIN Maximal-multipoint-Seshadri-constants-in-positive-characteristic//04-reshape.tex ====
\section{Reshaping two intercepts}\label{sec:reshape}
We now change two intercepts of a simplex while preserving their product.
The support sets need not fill the simplex: the construction applies to any
graded family inside it. Its algebraic step uses the two branches of a node;
its remaining steps are lattice changes and coordinate compressions.
This follows the planar construction of
\citet[Section~4, Proposition~4.1]{OpenAI2026Higher}, with a formal node
whose branches remain distinct in characteristic two. The nodal bound,
compressions, and primitive-direction choice also appear in
\citet[Propositions~3.2 and~3.4, Lemma~4.3]{OpenAI2026Surfaces};
we give all the arguments needed here.
\begin{proposition}[Two-intercept reshaping]\label{prop:reshape}
Let $n\ge2$, let $A,B,a_3,\ldots,a_n>0$, and let $(S_N)_{N\ge1}$ be a
graded family of finite subsets of $\Z^n$ satisfying
\[
S_N\subset N\Delta(A,B,a_3,\ldots,a_n).
\]
Put $H=AB$ and $d=\max\{3/A,2/B\}$. For every $w>0$ with $dw<1/16$,
there is a graded family $(S'_N)_{N\ge1}$ such that
\[
S'_N\subset N\Delta(w,H/w,a_3,\ldots,a_n),
\qquad \#S'_N=\#S_N \quad(N\ge1).
\]
The transformation from $S_N$ to $S'_N$ has backward jet transfer in every
degree. For $n=2$, the lists $a_3,\ldots,a_n$ are empty.
\end{proposition}
We first prove the nodal bound and then choose the lattice direction that
produces the prescribed intercept $w$. Throughout this section, $H$ and $d$
have the meanings above, and we put
\begin{equation}\label{eq:reshape-constants}
W=dH=\max\{2A,3B\},\qquad c=1/d,\qquad
T=d^2H=\max\{9B/A,4A/B\}\ge6.
\end{equation}
In particular, $cW=H$.
\subsection{The two branches of a node}
For $t>0$, order pairs $(\alpha,\beta)\in\Z_{\ge0}^2$ by the
lexicographic order of $(\alpha+t\beta,\beta)$, as in
Lemma~\ref{lem:exposing-weights}. We denote the least exponent in this
order by $\nu_t$.
\begin{lemma}[Nodal support bound]\label{lem:nodal-bound}
There are formal coordinates $\xi,\eta$ at the origin of $\A^2_k$ with
\[
y^2+xy-x^3=\xi\eta
\]
for which the following assertion holds. Given $t>0$, set
\[
a=\frac{Wt}{1+t},\qquad b=\frac{W}{1+t},\qquad
Q=\operatorname{conv}\{(0,0),(a,0),(c,c),(0,b)\}.
\]
If $\kappa\ge0$ and $0\ne f\in k[x,y]$ has all its exponents in
$\kappa\Delta(A,B)$, then its expansion in these coordinates satisfies
$\nu_t(f)\in\kappa Q$.
\end{lemma}
\begin{proof}
Write $g=y^2+xy-x^3$. The parametrization
\begin{equation}\label{eq:node-parametrization}
x=v(v+1),\qquad y=v^2(v+1)=vx
\end{equation}
identifies the complement of $x=0$ in $g=0$ with
$\operatorname{Spec}k[v,1/(v(v+1))]$: the inverse is $v=y/x$.
Consequently, a polynomial vanishes under \eqref{eq:node-parametrization}
if and only if it is divisible by $g$. Indeed, vanishing implies
membership in $(g)$ after inverting $x$, and $g$ is relatively prime to
$x$ in the unique factorization domain $k[x,y]$.
The polynomial $Z^2+Z-x$ has simple roots $0$ and $-1$ modulo $x$.
Hensel lifting gives roots $r_1,r_2\in k[[x]]$ with these respective
constant terms. Set
\[
\eta=y-xr_1(x),\qquad \xi=y-xr_2(x).
\]
Their linear terms are $y$ and $y+x$, so they are formal coordinates, and
$\xi\eta=g$. These assertions also hold in characteristic two: the roots
$0$ and $-1=1$ remain distinct and the linear terms remain independent.
Along \eqref{eq:node-parametrization} near $v=0$, uniqueness of the lifted
root gives $r_1(x(v))=v$, so $\eta=0$ and $\xi$ has order one in $v$.
Near $v=-1$, one similarly has $\xi=0$ and $\eta$ has order one in $v+1$.
Factor $f=g^e f_0$ with $e\ge0$ and $g\nmid f_0$. Give $x,y$ the
positive weights $1/A,1/B$, and write $\deg_{A,B}$ for the maximum weight
of a polynomial's monomials. This degree is additive on products,
because their highest weight forms multiply nontrivially. Moreover,
\[
\deg_{A,B}(g)=d,
\qquad \deg_{A,B}(f_0)\le\kappa-ed;
\]
here $1/A+1/B\le\max\{3/A,2/B\}=d$. Thus $ed\le\kappa$.
The nonzero polynomial
\[
F_0(v)=f_0\bigl(v(v+1),v^2(v+1)\bigr)
\]
has degree at most $W(\kappa-ed)$, since
\[
2i+3j\le W(i/A+j/B)
\]
for every monomial $x^iy^j$ of $f_0$. Its expansions at $v=0$ and
$v=-1$ are both nonzero, since $k[v]$ injects into each completed local
ring. These expansions are exactly the restrictions of $f_0$ to the
two formal branches described above.
Let $\nu_t(f_0)=(\alpha,\beta)$ and $\mu=\alpha+t\beta$.
Every term of its formal expansion has weight at least $\mu$.
Restricting to $\eta=0$ therefore gives order at least $\mu$ on the branch
at $v=0$; restricting to $\xi=0$ gives order at least $\mu/t$ on the
branch at $v=-1$. The nonzero polynomial $F_0$ consequently satisfies
\[
(1+1/t)\mu
\le \ord_{v=0}(F_0)+\ord_{v=-1}(F_0)
\le \deg(F_0)\le W(\kappa-ed).
\]
Equivalently, $(\alpha,\beta)\in(\kappa-ed)\Delta(a,b)$.
Because $g=\xi\eta$, multiplicativity of least exponents gives
\[
\nu_t(f)=(e,e)+\nu_t(f_0)
=ed(c,c)+\nu_t(f_0)\in\kappa Q.
\]
This also covers $\kappa=0$, when $f$ is constant.
\end{proof}
\subsection{A lattice direction of prescribed height}
Choose $t>T$. With $a,b$ as in Lemma~\ref{lem:nodal-bound}, one has
\[
a+b=W,\qquad
\frac ca+\frac cb=\frac{(1+t)^2}{Tt}>1.
\]
Thus $Q$ is a quadrilateral with the vertices in the order displayed in
that lemma, and its area is $c(a+b)/2=H/2$. The shear
$(\alpha,\beta)\mapsto(\alpha-\beta,\beta)$ sends its vertices to
\[
(0,0),\quad(a,0),\quad(0,c),\quad(-b,b).
\]
Its vertical slice at horizontal coordinate $q$ has length
\begin{equation}\label{eq:first-slice-length}
\begin{cases}
c(1+q/b),&-b\le q\le0,\\
c(1-q/a),&0\le q\le a.
\end{cases}
\end{equation}
Compression in the second coordinate therefore places the support in the
triangle
\begin{equation}\label{eq:reshape-triangle}
P=\operatorname{conv}\{(-b,0),(a,0),(0,c)\}.
\end{equation}
Its area is again $H/2$; see Figure~\ref{fig:reshape-polygons}.
\begin{figure}[tbp]
\centering
\begin{tikzpicture}[font=\small,>=stealth]
\begin{scope}[xscale=.9,yscale=2.4]
\filldraw[fill=blue!9,draw=blue!60!black,thick]
(0,0)--(4,0)--(.7,.7)--(0,.5)--cycle;
\node[below left] at (0,0) {$0$};
\node[below] at (4,0) {$(a,0)$};
\node[above] at (.7,.7) {$(c,c)$};
\node[left] at (0,.5) {$(0,b)$};
\node at (1.65,.22) {$Q$};
\end{scope}
\draw[->,thick] (4.05,.8)--(5.25,.8)
node[midway,above,align=center] {shear, then\\compress};
\begin{scope}[shift={(6.1,0)},xscale=.9,yscale=2.4]
\filldraw[fill=blue!9,draw=blue!60!black,thick]
(-.5,0)--(4,0)--(0,.7)--cycle;
\draw[densely dashed] (0,0)--(0,.7);
\node[below] at (-.5,0) {$(-b,0)$};
\node[below] at (4,0) {$(a,0)$};
\node[above] at (0,.7) {$(0,c)$};
\node at (1.6,.22) {$P$};
\end{scope}
\end{tikzpicture}
\caption{The nodal bound $Q$ becomes the triangle $P$ after a shear and
vertical compression. Each vertical slice of the sheared bounding polygon
is replaced by an interval of the same length starting at zero;
Formula~\eqref{eq:first-slice-length}
gives the resulting boundary. Both polygons have area $H/2$. The drawing
is schematic; the displayed vertex coordinates are exact.}
\label{fig:reshape-polygons}
\end{figure}
The next lemma chooses both $t$ and a primitive lattice direction. We
seek unimodular coordinates in which $P$ has vertical extent $H/w$;
its area $H/2$ will then force its largest horizontal slice to have
length $w$.
\begin{lemma}[Prescribed primitive height]\label{lem:primitive-height}
If $w>0$ and $dw<1/16$, there exist $t>T$ and a primitive vector
$u=(i,j)\in\Z^2$ such that the three vertex heights
$z\mapsto\det(u,z)$ of the triangle $P$ in
\eqref{eq:reshape-triangle} are distinct and their range has length $H/w$.
\end{lemma}
\begin{proof}
Put $R=H/w$ and $a_0=WT/(1+T)$. As $t$ ranges over $(T,\infty)$,
the value $a=Wt/(1+t)$ ranges over $(a_0,W)$.
For $u=(i,j)$, the heights of $(-b,0),(a,0),(0,c)$ are $jb,-ja,i/d$.
We seek $j>0$ and $i$ such that
\[
ja+i/d=R,\qquad i/d>jb.
\]
The first equality turns the second inequality into $R-jW>0$.
Thus we first choose $jW\le R/2$, and then use the freedom in $a$ to
obtain the equality with $i$ coprime to $j$.
Take $j$ to be the largest power of two at most
\[
\frac{R}{2W}=\frac{1}{2dw}.
\]
Then $j>1/(4dw)>4$. The open interval
\[
\bigl(d(R-jW),\ d(R-ja_0)\bigr)
\]
has length
\[
dj(W-a_0)=\frac{jT}{1+T}>2,
\]
so it contains an odd integer $i$. Since $j$ is a power of two, $(i,j)$
is primitive. There is a unique $t>T$ such that
\[
a=\frac{R-i/d}{j}\in(a_0,W).
\]
The resulting heights satisfy
\[
-ja<jb<i/d,\qquad
i/d-jb=R-jW\ge R/2>0,\qquad i/d+ja=R.
\]
This gives the required height range and distinctness.
\end{proof}
\subsection{Completion of the transformation}
\begin{proof}[Proof of Proposition~\ref{prop:reshape}]
Fix $t$ and $u$ from Lemma~\ref{lem:primitive-height}; these choices will
be used in every degree. Keep coordinates $3,\ldots,n$ unchanged.
For each degree $N$ and each label
$\gamma=(\gamma_3,\ldots,\gamma_n)$ occurring in $S_N$, put
\[
\kappa=N-\sum_{h=3}^n\frac{\gamma_h}{a_h}\ge0.
\]
The pair exponents with this label lie in $\kappa\Delta(A,B)$.
Take their polynomial span in $k[x,y]$ and replace it by its least
exponents in the coordinates $\xi,\eta$ and order $\nu_t$.
Lemma~\ref{lem:least-terms} preserves cardinality and gradedness: the row
spaces multiply into the row with degree and label added, and the same
coordinates and order are used throughout. For fixed $N$, choose a basis
with distinct least exponents in each row and apply
Lemma~\ref{lem:exposing-weights} to the finite union of these bases.
Its common weight vector allows Lemma~\ref{lem:formal-transfer} to apply
to all labels at once, giving backward jet transfer. The old points can
be chosen in the dense open torus of $\A^2\times\T^{n-2}$.
Lemma~\ref{lem:nodal-bound} places the new pair exponents in $\kappa Q$.
Apply the shear above and compress in the second coordinate.
Lemmas~\ref{lem:unimodular} and~\ref{lem:compression}, together with
\eqref{eq:first-slice-length}, give the bound $\kappa P$.
It remains to turn this triangle into $\Delta(w,H/w)$.
Complete the primitive vector $u$ to an integral basis $(u,v)$ with
$\det(u,v)=1$, and use the coordinates in that basis. This is a
unimodular change, and its second coordinate is precisely
$z\mapsto\det(u,z)$. Write $q_0<q_1<q_2$ for the three vertex heights of
the transformed triangle. Its horizontal slice length is linear from
zero at $q_0$ to a maximum at $q_1$, and linear back to zero at $q_2$.
The maximum is $w$, since area is preserved and
\[
\frac H2=\frac12(q_2-q_0)\,\bigl(\text{maximum slice length}\bigr),
\qquad q_2-q_0=R=H/w.
\]
Compressing in the first coordinate therefore gives the bound
\[
\operatorname{conv}\{(0,q_0),(w,q_1),(0,q_2)\}.
\]
Its vertical slice at first coordinate $h\in[0,w]$ has length
$R(1-h/w)$. Compression in the second coordinate gives
$\Delta(w,R)$, as illustrated in Figure~\ref{fig:final-compression}. These slice statements commute with multiplication by
$\kappa$, so the resulting pair bound is $\kappa\Delta(w,R)$ for every
label. If $\kappa=0$, its only possible pair exponent is $(0,0)$, which
remains unchanged at every step.
Restoring the labels gives
$S'_N\subset N\Delta(w,R,a_3,\ldots,a_n)$. Every step preserves
cardinality and gradedness and has backward jet transfer, so their
composition has all the asserted properties.
\end{proof}
\begin{figure}[tbp]
\centering
\begin{tikzpicture}[font=\small,>=stealth]
\begin{scope}
\filldraw[fill=blue!9,draw=blue!60!black,thick]
(0,-.6)--(2,.4)--(0,1.8)--cycle;
\draw[densely dashed] (1,-.1)--(1,1.1);
\node[left] at (0,-.6) {$(0,q_0)$};
\node[left] at (0,1.8) {$(0,q_2)$};
\node[right] at (2,.4) {$(w,q_1)$};
\end{scope}
\draw[->,thick] (3.65,.6)--(5.15,.6)
node[midway,above,align=center] {vertical\\compression};
\begin{scope}[shift={(6.5,-.6)}]
\filldraw[fill=blue!9,draw=blue!60!black,thick]
(0,0)--(2,0)--(0,2.4)--cycle;
\draw[densely dashed] (1,0)--(1,1.2);
\node[below left] at (0,0) {$0$};
\node[below] at (2,0) {$(w,0)$};
\node[left] at (0,2.4) {$(0,R)$};
\end{scope}
\end{tikzpicture}
\caption{After the lattice change and horizontal compression, a final
vertical compression produces the target triangle. At first coordinate
$h$, both vertical slices have length $R(1-h/w)$, where
$R=q_2-q_0=H/w$. Dashed segments mark corresponding slices.}
\label{fig:final-compression}
\end{figure}
==== END Maximal-multipoint-Seshadri-constants-in-positive-characteristic//04-reshape.tex ====
==== BEGIN Maximal-multipoint-Seshadri-constants-in-positive-characteristic//05-interpolation.tex ====
\section{Interior filling and interpolation}\label{sec:interpolation}
The reshaping construction preserves the number of exponents, but does not
identify the individual exponents that survive. Gradedness supplies the missing
information: a family with full asymptotic density contains every lattice point
in a fixed compact subset of the interior, in all sufficiently large degrees.
We first prove this statement, then use it to find a complete interpolation
space inside the reshaped supports. These two arguments follow
\cite[Lemmas~5.1 and~5.2]{OpenAI2026Higher}. Interior filling for semigroups
is studied more generally in \cite[Theorem~1.6]{KavehKhovanskii2012}; the
full-density hypothesis here permits a direct proof by counting decompositions.
\begin{lemma}[Interior filling]\label{lem:interior-filling}
Let $P\subset\R^n$, $n\ge1$, be a compact convex set with nonempty interior.
Suppose finite sets $F_N\subset NP\cap\Z^n$, indexed by integers $N\ge1$,
satisfy
\[
F_N+F_{N'}\subset F_{N+N'}
\quad\text{and}\quad
\#F_N=\vol(P)N^n+o(N^n).
\]
Then for every compact set $K\subset\operatorname{int}(P)$ there is an integer
$N_K$ such that
\[
NK\cap\Z^n\subset F_N\qquad(N\ge N_K).
\]
\end{lemma}
\begin{proof}
The boundary of a convex body has measure zero, so lattice-point counting by
Riemann sums gives
\[
\#(NP\cap\Z^n)=\vol(P)N^n+o(N^n).
\]
Consequently the number of missing lattice points
\[
b_N:=\#\bigl((NP\cap\Z^n)\setminus F_N\bigr)
\]
is $o(N^n)$. We show that a point of $NK$ has too many decompositions into two
points in nearly equal degrees for all of them to involve a missing point.
Assume $K$ is nonempty, and choose $\rho>0$ so that the closed ball of radius
$\rho$ about every point of $K$ lies in $P$. Write
$N=N_1+N_2$ with $N_1=\lfloor N/2\rfloor$ and
$N_2=\lceil N/2\rceil$. For $y\in NK\cap\Z^n$, consider all lattice points
\[
y_1\in\Z^n,
\qquad
\left\|y_1-\frac{N_1}{N}y\right\|<\rho N_1,
\qquad y_2:=y-y_1.
\]
Both $y_1/N_1$ and $y_2/N_2$ lie in the radius-$\rho$ ball about $y/N$.
Thus $y_i\in N_iP\cap\Z^n$. The ball from which $y_1$ is chosen contains
at least $cN^n$ lattice points for a constant $c>0$ independent of $y$ and
all sufficiently large $N$: an inscribed cube gives this bound uniformly in
the center. Among these choices, at most $b_{N_1}$ have $y_1\notin F_{N_1}$,
and at most $b_{N_2}$ have $y_2\notin F_{N_2}$, since $y_1\mapsto y-y_1$ is
injective. As $b_{N_1}+b_{N_2}=o(N^n)$, at least one choice has
$y_i\in F_{N_i}$. Gradedness then gives $y=y_1+y_2\in F_N$.
All bounds are uniform in $y$, which proves the assertion.
\end{proof}
The interpolation space we need is a simplex stretched in the last coordinate
by the number of points. Its fibers over the other exponents have exactly the
lengths needed for one-variable interpolation with multiplicities.
\begin{lemma}[Jets from an elongated simplex]\label{lem:elongated-jets}
Let $n,r\ge1$ and $m\ge0$ be integers, and let $k$ be a field containing
distinct nonzero elements $c_1,\ldots,c_r$. Set
\begin{equation}\label{eq:interpolation-support}
E_m=\left\{(\beta,b)\in\Z_{\ge0}^{n-1}\times\Z_{\ge0}:
|\beta|+\frac br<m+1\right\},
\qquad |\beta|=\sum_{j=1}^{n-1}\beta_j.
\end{equation}
Then $M(E_m)$ separates jets through order $m$ at the $r$ points
\[
q_i=(1,\ldots,1,c_i)\in\T^n(k).
\]
For $n=1$, the tuple $\beta$ is empty and $|\beta|=0$.
\end{lemma}
\begin{proof}
Write the coordinates as $(x_1,\ldots,x_{n-1},Y)$ and put
$x'-1=(x_1-1,\ldots,x_{n-1}-1)$. An arbitrary jet through order $m$ at $q_i$
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 gives a polynomial
$h_\alpha(Y)$ satisfying
\[
h_\alpha(Y)\equiv h_{i,\alpha}(Y-c_i)
\pmod{(Y-c_i)^{m-|\alpha|+1}}
\quad(1\le i\le r),
\qquad
\deg h_\alpha<r(m-|\alpha|+1).
\]
Indeed, the ideals in this display are pairwise comaximal, and their product
is generated by a monic polynomial of degree $r(m-|\alpha|+1)$.
The polynomial
\[
\sum_{|\alpha|\le m}(x'-1)^\alpha h_\alpha(Y)
\]
realizes the prescribed jets. Each monomial $x'^\beta Y^b$ in its expansion
satisfies $\beta_j\le\alpha_j$ for every $j$ and
$b<r(m-|\alpha|+1)$ for some $\alpha$. Hence
$|\beta|+b/r<m+1$, so the polynomial belongs to $M(E_m)$.
This uses powers of local parameters and the Chinese remainder theorem,
and therefore works in every characteristic.
\end{proof}
We now choose the number of points large enough that successive applications
of Proposition~\ref{prop:reshape} produce the required elongated simplex.
\begin{proposition}[Jets at the geometric generic tuple]\label{prop:generic-jets}
Let $k$ be an algebraically closed field of positive characteristic, let $X/k$
be a smooth integral projective variety of dimension $n\ge2$, and let $L$ be
ample. Put $V=L^n$. There is an integer $r_0=r_0(X,L)$ such that, for every
integer $r\ge r_0$ and every real number
\[
0<\theta<\left(\frac Vr\right)^{1/n},
\]
there is an integer $N_0=N_0(r,\theta)$ such that, for every $N\ge N_0$,
the sections of $NL$ separate jets through order $\lfloor N\theta\rfloor$
at some ordered $r$-tuple of distinct $k$-points, which may depend on $N$.
For each such $N$, the same jet separation holds after base change to $K_r$
at the geometric generic tuple $p_1,\ldots,p_r$.
\end{proposition}
\begin{proof}
Take the graded family $S_N$ and simplex
$P_0=\Delta(a_1,\ldots,a_n)$ from Proposition~\ref{prop:initial-simplex}.
Thus $\prod_j a_j=V$, the support $S_N$ lies in $NP_0$, and
\[
\#S_N=\frac{V}{n!}N^n+o(N^n).
\]
Let $\ell=\min_j a_j$, and choose $r_0$ so large that
\[
w:=\left(\frac Vr\right)^{1/n}<\frac{\ell}{100}
\qquad(r\ge r_0).
\]
Fix one such $r$. Reshape the first two intercepts to $(w,a_1a_2/w)$,
then reshape the second and third intercepts, and continue through the
$(n-1)$st and $n$th intercepts. At every step, both intercepts $A,B$
being reshaped are at least $\ell$: untouched intercepts have this property,
and the carried intercept $AB/w$ is larger than $A$ because $B>w$.
Consequently
\[
w\max(3/A,2/B)\le\frac{3w}{\ell}<\frac1{16},
\]
as required by Proposition~\ref{prop:reshape}. The product of all intercepts
remains $V$. The resulting graded family $F_N$ therefore satisfies
\begin{equation}\label{eq:elongated-full-density}
\begin{aligned}
&F_N\subset NP_*\cap\Z^n,
\qquad P_*:=\Delta(w,\ldots,w,rw),\\
&\#F_N=\frac{V}{n!}N^n+o(N^n)
=\vol(P_*)N^n+o(N^n).
\end{aligned}
\end{equation}
Here the last intercept is $V/w^{n-1}=rw$.
The transformations preserve backward jet transfer, so jet separation by
$M(F_N)$ will imply jet separation by $H^0(X,NL)$.
Fix $0<\theta<w$. To apply Lemma~\ref{lem:interior-filling}, we contract
$P_*$ and then translate it away from the coordinate faces.
Choose $\lambda$ with $\theta/w<\lambda<1$, and choose a vector
$v\in\R_{>0}^n$ small enough that
\[
v+\lambda P_*\subset\operatorname{int}(P_*).
\]
For example, it suffices that
$\sum_{j<n}v_j/w+v_n/(rw)<1-\lambda$.
Choose vectors $v_N\in N^{-1}\Z^n$ tending to $v$.
For all sufficiently large $N$, the sets $v_N+\lambda P_*$ lie in a common
compact subset $K$ of $\operatorname{int}(P_*)$. Applying
Lemma~\ref{lem:interior-filling} to \eqref{eq:elongated-full-density} gives
\begin{equation}\label{eq:translated-inner-support}
Nv_N+\bigl(N\lambda P_*\cap\Z^n\bigr)\subset F_N
\end{equation}
for all sufficiently large $N$.
Put $m=\lfloor N\theta\rfloor$. Since $\theta<\lambda w$, we have
$m+1\le N\lambda w$ for all sufficiently large $N$.
The defining inequality \eqref{eq:interpolation-support} therefore gives
\[
E_m\subset N\lambda P_*\cap\Z^n.
\]
Choose distinct $c_1,\ldots,c_r\in k^\times$ and apply
Lemma~\ref{lem:elongated-jets}. Multiplication by the Laurent monomial with
exponent $Nv_N\in\Z^n$ is multiplication by a unit on the torus, hence
induces an automorphism on every jet space. Thus
\eqref{eq:translated-inner-support} implies that $M(F_N)$ separates jets
through order $m$ at these $r$ points. Backward jet transfer through all
reshapings and Proposition~\ref{prop:initial-simplex} gives the asserted
jet separation at an ordered tuple on $X$.
For each fixed $r$, $\theta$, and sufficiently large $N$, the locus in $U_r$
where this evaluation map is surjective is a nonempty open set, by the
openness of jet separation established in Section~\ref{sec:transfer}.
The variety $U_r$ is integral, so its generic point belongs to that open set.
Base change of the jet evaluation map to $K_r$ preserves surjectivity and
identifies its targets with the jets at $p_1,\ldots,p_r$. The same
geometric generic tuple therefore satisfies every required jet-separation
statement.
\end{proof}
==== END Maximal-multipoint-Seshadri-constants-in-positive-characteristic//05-interpolation.tex ====
==== BEGIN Maximal-multipoint-Seshadri-constants-in-positive-characteristic//06-positivity.tex ====
\section{From jets to the Seshadri constant}\label{sec:positivity}
The interpolation statement now gives the lower bound for the Seshadri
constant. We first spell out the passage from jets to intersection numbers,
including at singular points of a curve, and then recall the matching volume
bound. The relation between asymptotic jet separation and Seshadri constants
is classical; see \cite[Theorem~6.4]{Demailly1992} in the complex setting
and \cite[Section~2, Equation~(2.3)]{MustataSchwede2014} in arbitrary
characteristic.
The arguments below work in arbitrary characteristic.
\begin{lemma}[Jets and curve multiplicities]\label{lem:jet-curve-bound}
Let $Y$ be a smooth integral projective variety over an algebraically closed
field, let $A$ be a line bundle on $Y$, and let $q_1,\ldots,q_r$ be distinct
points. Suppose that $H^0(Y,A)$ separates jets through order $m\geq 1$ at
these points. Then every integral curve $C\subset Y$ meeting at least one
of them satisfies
\[
A\cdot C\geq m\sum_{i=1}^r\mult_{q_i}C,
\]
where the multiplicity is zero at a point outside $C$.
\end{lemma}
\begin{proof}
Choose such a point $q_j$. Write $R=\mathcal O_{C,q_j}$ and let $\mathfrak n$
be its maximal ideal. The vector space
$\mathfrak n^m/\mathfrak n^{m+1}$ is nonzero: otherwise Nakayama's lemma would
give $\mathfrak n^m=0$, contrary to $R$ being a one-dimensional local domain.
In a local frame of $A$, lift a nonzero class in this vector space to an
ambient jet of order $m$ at $q_j$. This lift can be chosen in the image of
$\mathfrak m_{Y,q_j}^m$. Prescribe that jet at $q_j$ and the zero jet at all
other marked points. Jet separation produces a section $\sigma\in H^0(Y,A)$
which vanishes to order at least $m$ at every $q_i$ and whose restriction to
$C$ is nonzero.
For completeness, the local multiplicity estimate used here is
\begin{equation}\label{eq:local-zero-length}
\operatorname{length}_R(R/(f))\geq m\,e(R)
\qquad(0\ne f\in\mathfrak n^m),
\end{equation}
where $e(R)$ denotes the Hilbert--Samuel multiplicity of a one-dimensional
Noetherian local domain $(R,\mathfrak n)$; see
\cite[Definition~43.15.1, Tag~0AZV]{StacksProject}. Indeed, for positive integers $t$,
the nonzero divisor $f$ gives
\[
t\operatorname{length}_R(R/(f))
=\operatorname{length}_R(R/(f^t))
\geq\operatorname{length}_R(R/\mathfrak n^{mt})
=mt\,e(R)+O(1).
\]
Divide by $t$ and let $t$ tend to infinity. Applying
\eqref{eq:local-zero-length} to the local equations of $\sigma|_C$ at the
marked points proves the inequality: the sum of all local zero lengths of a
nonzero section of $A|_C$ is $\deg(A|_C)=A\cdot C$.
\end{proof}
\begin{lemma}[Volume bound]\label{lem:volume-bound}
Let $Y$ be a smooth integral projective variety of dimension $d\geq2$ over an
algebraically closed field, let $A$ be ample, and let $q_1,\ldots,q_r$ be
distinct points. Then
\[
\inf_{C\cap\{q_1,\ldots,q_r\}\ne\varnothing}
\frac{A\cdot C}{\sum_i\mult_{q_i}C}
\leq \left(\frac{A^d}{r}\right)^{1/d},
\]
where the infimum is over integral curves in $Y$.
\end{lemma}
\begin{proof}
Denote the infimum by $\varepsilon$. If $\varepsilon=0$, the conclusion
is immediate. Otherwise, blow up the marked points:
\[
\pi:\widetilde Y\longrightarrow Y,
\qquad E_1,\ldots,E_r\subset\widetilde Y.
\]
For each rational number $0\leq s<\varepsilon$, the divisor class
\[
D_s=\pi^*A-s\sum_i E_i
\]
is nef. To check this on curves, first observe that
$D_s|_{E_i}=s\mathcal O_{\PP^{d-1}}(1)$, which has nonnegative degree on every
curve in $E_i$. Every other integral curve in $\widetilde Y$ is the strict
transform $\widetilde C$ of an integral curve $C\subset Y$, and
\begin{equation}\label{eq:exceptional-multiplicity}
E_i\cdot\widetilde C=\mult_{q_i}C.
\end{equation}
Thus $D_s\cdot\widetilde C\geq0$ by the definition of $\varepsilon$; for curves
missing all marked points this follows from the ampleness of $A$.
Here is a local justification of \eqref{eq:exceptional-multiplicity}, which
also covers singular curves. If $q_i\in C$, the Rees algebra construction
identifies $\widetilde C$ with the blow-up of $C$ at the marked points.
Indeed, the ambient Rees algebra surjects onto the Rees algebra of the
restricted ideal. The latter is integral, and its Proj agrees with $C$
away from the centers, so its closed image is exactly the strict transform.
For
$R=\mathcal O_{C,q_i}$ with maximal ideal $\mathfrak n$, its exceptional fiber
is
\[
\operatorname{Proj}\bigl(\operatorname{gr}_{\mathfrak n}R\bigr).
\]
It is the effective Cartier divisor cut out on $\widetilde C$ by $E_i$.
Its length is $e(R)$: the Hilbert function
$\dim_{k(q_i)}(\mathfrak n^t/\mathfrak n^{t+1})$ equals $e(R)$ for all
sufficiently large $t$, and its eventual value is the length of this
zero-dimensional projective scheme. This proves
\eqref{eq:exceptional-multiplicity}; if $q_i\notin C$, both sides vanish.
A nef class on a projective variety is a limit of ample classes and therefore
has nonnegative top self-intersection. The exceptional divisors are
disjoint, $\pi^*A$ restricts trivially to each of them, and
$E_i^d=(-1)^{d-1}$. Consequently
\[
0\leq D_s^d=A^d-rs^d.
\]
If $\varepsilon>(A^d/r)^{1/d}$, even if $\varepsilon=+\infty$, there is a
rational number $s$ strictly between $(A^d/r)^{1/d}$ and $\varepsilon$.
The displayed inequality excludes such an $s$, proving the bound.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:main}]
Choose $r_0$ as in Proposition~\ref{prop:generic-jets} and fix $r\geq r_0$.
Put $V=L^n$ and $w=(V/r)^{1/n}$. Work over $K_r$ at the geometric generic tuple
$p_1,\ldots,p_r$. For every fixed real number $0<\theta<w$ and all
sufficiently large integers $N$, Proposition~\ref{prop:generic-jets} gives
separation of jets through order $m=\lfloor N\theta\rfloor$ by
$H^0(X_{K_r},L_{K_r}^{\otimes N})$ at this fixed tuple.
For large $N$ we have $m\geq1$, so Lemma~\ref{lem:jet-curve-bound} yields
\[
\frac{L_{K_r}\cdot C}{\sum_i\mult_{p_i}C}
\geq\frac{\lfloor N\theta\rfloor}{N}
\]
for every integral curve in the defining infimum. Letting $N$ tend to
infinity gives $\varepsilon_r(X,L)\geq\theta$. Since $\theta<w$ was
arbitrary, $\varepsilon_r(X,L)\geq w$. Intersection numbers are unchanged
by extension of the algebraically closed ground field, so
$(L_{K_r})^n=V$; Lemma~\ref{lem:volume-bound} supplies the reverse inequality.
\end{proof}
\begin{remark}[Surfaces]\label{rem:surfaces}
The same proof gives the conclusion of Theorem~\ref{thm:main} when $n=2$.
Indeed, Propositions~\ref{prop:initial-simplex}, \ref{prop:reshape},
and~\ref{prop:generic-jets} all allow $n=2$; the reshaping is then performed
just once, with no unchanged coordinates. Both intersection lemmas above
also apply in dimension two.
\end{remark}
==== END Maximal-multipoint-Seshadri-constants-in-positive-characteristic//06-positivity.tex ====
==== BEGIN Maximal-multipoint-Seshadri-constants-in-positive-characteristic//main.tex ====
\documentclass[11pt]{article}
\input{preamble}
\title{Maximal multipoint Seshadri constants\\in positive characteristic}
\author{OpenAI}
\date{October 5, 2026}
\begin{document}
\maketitle
\begin{abstract}
Let $L$ be an ample line bundle on a smooth integral projective variety $X$
over an algebraically closed field of positive characteristic, with
$\dim X=n\ge3$. 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
the geometric generic tuple of $r$ points equals the volume bound
$(L^n/r)^{1/n}$. The same conclusion holds in dimension two. This establishes
the positive-characteristic form of the qualitative Nagata--Biran--Szemberg
assertion at geometric generic tuples.
\end{abstract}
\input{sections/01-introduction}
\input{sections/02-transfer}
\input{sections/03-flag}
\input{sections/04-reshape}
\input{sections/05-interpolation}
\input{sections/06-positivity}
\bibliographystyle{plainnat}
\begingroup\raggedright\interlinepenalty=10000
\bibliography{references/references}
\endgroup
\end{document}
==== END Maximal-multipoint-Seshadri-constants-in-positive-characteristic//main.tex ====
==== BEGIN Maximal-multipoint-Seshadri-constants-in-positive-characteristic//preamble.tex ====
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==== BEGIN Maximal-multipoint-Seshadri-constants-in-positive-characteristic//references.bib ====
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