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Math-2564 · mathematics · Algebraic geometry

Generic maximal Seshadri constants in positive characteristic

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Over any algebraically closed field of positive characteristic, for a smooth projective geometrically integral n-fold X with $n\geq 3$ and ample L, does some $r_{0}(X,L)$ guarantee $\varepsilon (L;x_{1},\dots ,x_r)=(L^n/r)^{1/n}$ at the geometric generic ordered tuple for every $r\geq r_{0}$?

Short English statement

Over any algebraically closed field of positive characteristic, for a smooth projective geometrically integral n-fold X with $n\geq 3$ and ample L, does some $r_{0}(X,L)$ guarantee $\varepsilon (L;x_{1},\dots ,x_r)=(L^n/r)^{1/n}$ at the geometric generic ordered tuple for every $r\geq r_{0}$?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2564-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2564

Manuscript 1 (family 039, 2026-10-05): The problem selects the mathematical statement summarized here, including its stated hypotheses and parameter range. The cited theorem gives a claimed proof of that assertion in full.

Generic maximal Seshadri constants in positive characteristic

OpenAI's claim

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. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
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.

Outline of the argument

Manuscript 1 (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.

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.

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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