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Math-2360 · mathematics · Algebraic geometry

Multipoint Seshadri Constants and Nagata-Type Conjectures

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Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
For sufficiently many sufficiently general points $Z$ on a polarized variety $(X,L)$, is the Seshadri constant $\varepsilon(X,L;Z)$ maximal possible? In $\mathbb{P}^{2}$ this includes Nagata's conjecture for $r\geq10$ general points and the SHGH prediction for plane linear systems.

Formulation references: [S1], [S2].

Short English statement

For sufficiently many sufficiently general points $Z$ on a polarized variety $(X,L)$, is the Seshadri constant $\varepsilon(X,L;Z)$ maximal possible? In $\mathbb{P}^{2}$ this includes Nagata's conjecture for $r\geq10$ general points and the SHGH prediction for plane linear systems.

Sources

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

Math-2360-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2360

Manuscript 1 (family 039, 2026-10-05): Maximal multipoint Seshadri constants on complex polarized varieties in dimensions at least three for sufficiently many points; the problem bundle additionally asks general polarized-variety Seshadri questions and SHGH formulas. The stated Nagata-type branch does not complete those other assertions.

Manuscript 2 (family 039, 2026-09-23): Maximal multipoint Seshadri constants on polarized complex surfaces for sufficiently many very general points; the problem bundle additionally asks general polarized-variety Seshadri questions and SHGH formulas. The stated Nagata-type branch does not complete those other assertions.

Manuscript 3 (family 039, 2026-10-05): Maximal multipoint Seshadri constants in positive characteristic and dimension at least three at the geometric generic tuple; the problem bundle additionally asks general polarized-variety Seshadri questions and SHGH formulas. The stated Nagata-type branch does not complete those other assertions.

Manuscript 4 (family 039, 2026-09-23): Nagata’s strict unequal-multiplicity inequality for every effective plane curve at r at least ten very general complex points; the problem bundle additionally asks general polarized-variety Seshadri questions and SHGH formulas. The stated Nagata-type branch does not complete those other assertions.

Multipoint Seshadri Constants and Nagata-Type Conjectures

OpenAI's claim

Manuscript 1 (family 039, 2026-10-05): In every dimension at least three, the ordinary multipoint Seshadri constant is claimed to reach $(L^n/r)^{1/n}$ at very general points for every sufficiently large $r$; Manuscript 2 (family 039, 2026-09-23): For any polarized smooth complex projective surface, the equal-weight multipoint Seshadri constant is claimed to reach $\sqrt{L^2/r}$ for all integers beyond a polarization-dependent threshold. Manuscript 3 (family 039, 2026-10-05): The same eventual volume-bound equality for ordinary multipoint Seshadri constants is claimed in positive characteristic at geometric generic tuples, in dimensions two and higher; Manuscript 4 (family 039, 2026-09-23): At $r\ge10$ very general complex plane points, every effective curve is claimed to have total assigned multiplicity below its degree times $\sqrt r$, including reducible curves and unequal assignments.

Mathematical statement of the claimed result

Manuscript 1.
Let $X$ be smooth integral projective over $\mathbb C$, with $n=\dim X\ge3$, and $L$ ample. OpenAI claims an integer $r_0(X,L)$ such that for every $r\ge r_0$ and very general ordered distinct $p_1,\ldots,p_r$, \[ \varepsilon(X,L;\mathbf p)=\left(\frac{L^n}{r}\right)^{1/n}. \] The multipoint Seshadri constant is the supremum of $t\ge0$ for which $\pi^*L-t\sum_iE_i$ is nef on the blow-up at these points. For each $r$, the exceptional tuples lie in a union, indexed by a countable set, of proper algebraic closed subsets with nonempty complement. The threshold depends on $(X,L)$ and is not specified uniformly.
Manuscript 2.
For any smooth integral projective complex surface $S$ and ample line bundle $L$, OpenAI claims $r_0(S,L)$ such that every $r\ge r_0$ and very general tuple of distinct points give, on the blow-up $\pi$ with exceptional curves $E_i$, \[ \pi^*L-\sqrt{L^2/r}\sum_{i=1}^rE_i\text{ nef},\qquad \varepsilon(S,L;\mathbf p)=\sqrt{L^2/r}. \] The constant is the supremum of $t$ making $\pi^*L-t\sum E_i$ nef. Very general excludes countably many proper closed subsets of the distinct-point configuration space; the statement is for sufficiently many points, not every prescribed small $r$.
Manuscript 3.
Let $X/k$ be smooth integral projective over a field $k$ assumed algebraically closed, with characteristic $p>0$, with dimension $n\ge3$ and ample $L$. For $r$ ordered generic distinct points over the algebraic closure $K_r$ of the configuration-space function field, define \[ \varepsilon_r(X,L)=\inf_C\frac{L_{K_r}\cdot C}{\sum_{i=1}^r\operatorname{mult}_{p_i}C}, \] over integral curves meeting at least one point. OpenAI claims a threshold $r_0(X,L)$ with $\varepsilon_r(X,L)=(L^n/r)^{1/n}$ for every integer $r\ge r_0$. This is a geometric-generic-point statement, which avoids interpreting a countable exceptional-set complement over a countable field.
Manuscript 4.
For each $r\ge10$, OpenAI claims a nonempty very-general locus of ordered distinct points $(p_1,\ldots,p_r)$ in $\mathbb P^2_{\mathbb C}$ such that every nonzero effective plane curve $C$ of degree $d\ge1$ and integers $m_i\ge0$ with $\operatorname{mult}_{p_i}C\ge m_i$ satisfy \[ d\sqrt r>\sum_{i=1}^r m_i. \] Very general means excluding a union, indexed by a countable set, of proper algebraic closed subsets of the distinct-point configuration space. Multiplicities may be unequal, curves may be reducible, and the strict inequality starts at ten points.

Outline of the argument

Manuscript 1 (family 039, 2026-10-05): Graded exponent supports are reshaped into a volume-preserving simplex compatible with section multiplication. Full density forces every sufficiently deep interior lattice point to occur, allowing torus interpolation to separate the needed jets; very-general tuples inherit all the finite tests. Manuscript 2 (family 039, 2026-09-23): Finite jet tests are transferred to an algebraic torus through nodal-curve degeneration. A varying exponent region is tuned to a primitive lattice direction and a subgroup with exactly $r$ points, giving injectivity for all required orders rather than only a fixed multiplicity.

Manuscript 3 (family 039, 2026-10-05): Formal substitutions replace the analytic degeneration used in the complex companions. A nodal equation with two distinct branches even in characteristic two permits the graded-support reshaping and finite torus interpolation to proceed over the ground field. Manuscript 4 (family 039, 2026-09-23): Incidence loci and cyclic products reduce a hypothetical failure to equal multiplicities at arbitrary configurations. Degeneration toward a cubic produces finite jet constraints; colliding the remaining points and taking limits of elliptic theta bases turns the contradiction into a full-rank polynomial evaluation problem.

Formal verification

For family 039, the scope page documents the following selections. The selected plane-curve inequality allows unequal multiplicities and repeated components at every allowed point count. A separate statement gives eventual maximality on each polarized complex surface, with a surface-dependent threshold; the higher-dimensional and positive-characteristic companions are not selected here. Manuscript 1 is not among the papers selected by that page, so the family link does not formalize its main conclusion. The documented statement covers the main claim of manuscript 2. Manuscript 3 is not among the papers selected by that page, so the family link does not formalize its main conclusion. The documented statement covers the main claim of manuscript 4. Comparator files: MaximalSeshadriConstants.lean, Nagata.lean. Declarations: OAI.MaximalSeshadri.Geometry.maximalSeshadriConstants, OAI.Nagata.nagata_conjecture.

This site has not executed the Lean files or checked their dependencies. These descriptions report the scope of the linked declarations, rather than a new verification of the proofs.

Status caveat

OpenAI attributes these results to an internal model. Its release notice cautions that results lacking full formalization can contain mistakes. This site has not independently assessed the proofs; a solved label records the scope of the reported claim.

Sources

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