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Math-2238 · mathematics · Algebraic geometry

Wikipedia geometry item 14: Nagata–Biran conjecture that if $X$ is a smooth algebraic surface and $L$ is an ample line bund…

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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.
Nagata–Biran conjecture that if $X$ is a smooth algebraic surface and $L$ is an ample line bundle on $X$ of degree $d$, then for sufficiently large $r$, the Seshadri constant satisfies $\varepsilon(p_1,\ldots,p_r;X,L) = d/\sqrt{r}$.

Formulation references: [S1], [S2].

Short English statement

Nagata–Biran conjecture that if $X$ is a smooth algebraic surface and $L$ is an ample line bundle on $X$ of degree $d$, then for sufficiently large $r$, the Seshadri constant satisfies $\varepsilon(p_1,\ldots,p_r;X,L) = d/\sqrt{r}$.

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2238-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2238

Manuscript 1 (family 039, 2026-10-05): The embedded surface companion gives sqrt(L${}^{2}$/r) for all sufficiently large r. The catalogue formula uses an unspecified “degree d” as d/sqrt(r); no normalization equating d with sqrt(L${}^{2}$) is established, so exact full equivalence is withheld.

Manuscript 2 (family 039, 2026-09-23): The paper proves maximal multipoint constants on every smooth polarized complex surface beyond a surface-dependent threshold. The fragment writes a degree-based constant without a precise definition, so equivalence with the paper's sqrt(L squared/r) formulation is not safely established.

Wikipedia geometry item 14: Nagata–Biran conjecture that if \$X\$ is a smooth algebraic surface and \$L\$ is an ample line bund…

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.

Mathematical statement of the claimed result

Mathematical claim 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.
Mathematical claim 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 $r$.

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.

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. 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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