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Math-2549 · mathematics · Algebraic geometry

The generalized Mukai pseudoindex assertion

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For every smooth connected complex projective Fano $n$-fold $X$ of positive dimension, with Picard number $\rho$ and pseudoindex \[ \iota=\min\{-K_X\cdot C: C\text{ is a rational curve}\}, \] must $\rho(\iota-1)\leq n$? Must equality characterize the product of $\rho$ copies of $\mathbb{P}^{\iota-1}$?

Short English statement

For every smooth connected complex projective Fano $n$-fold $X$ with $n>0$, Picard number $\rho$, and pseudoindex $\iota=\min\{-K_X\cdot C:C\text{ is a rational curve}\}$, is $\rho(\iota-1)\leq n$, with equality exactly when $X$ is the product of $\rho$ copies of $\mathbb{P}^{\iota-1}$?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2549-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2549

Manuscript 1 (family 063, 2026-09-24): 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.

The generalized Mukai pseudoindex assertion

OpenAI's claim

For a smooth complex Fano $n$-fold, the generalized Mukai bound $\rho(\iota-1)\le n$ is claimed, with equality exactly for the corresponding product of projective spaces. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
For a smooth connected projective complex Fano manifold $X$ of dimension $n>0$, let $\rho_X$ be its Picard number and $\iota_X=\min\{-K_X\cdot C:C\text{ an irreducible rational curve}\}$ its pseudoindex. OpenAI claims \[ \rho_X(\iota_X-1)\le n, \qquad\text{equality}\ \Longleftrightarrow\ X\simeq(\mathbb P^{\iota_X-1})^{\rho_X}. \] The pseudoindex is a minimum curve degree, not divisibility of the anticanonical line bundle. There is no dimension restriction.

Outline of the argument

Manuscript 1 (family 063, 2026-09-24): A weighted-projective evaluation construction gives positive lower bounds for point descendants of free rational curves. Basis-invariant alternating traces detect independent curve degrees in quantum correspondences; minimal proper covering families enter in the equality case and identify the projective-space factors.

Formal verification

Family 063 has no Lean scope page in this snapshot; no formalization of these manuscript results is listed.

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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