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Math-2264 · mathematics · Algebraic geometry

A transcendental-rank obstruction to surface hosts

Status Partially solved · unsolvedmath.comLLM status Open unverified

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 a very general cubic fourfold $X$, is there a surface $S$ such that the Kuznetsov component $Ku(X) \subset D(X)$ has an embedding $Ku(X) \hookrightarrow D(S)$?

Formulation references: [S1], [S2].

Short English statement

For a very general cubic fourfold $X$, is there a surface $S$ such that the Kuznetsov component $Ku(X) \subset D(X)$ has an embedding $Ku(X) \hookrightarrow D(S)$?

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2264-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2264

The question concerns a very general cubic in the entire parameter space. The manuscript places its cubics on specified Hassett divisors and obtains a K3 derived-category equivalence there. Each such divisor is a proper special locus, so its examples do not belong to the very-general locus of the full parameter space. This is related categorical information, rather than a partial answer to the recorded generic surface-host question.

A transcendental-rank obstruction to surface hosts

OpenAI's claim

Very general cubic fourfolds in sufficiently large admissible Hassett divisors are claimed to be irrational even though their Kuznetsov categories come from ordinary projective K3 surfaces. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
An integer $d$ is called admissible when $d>6$, $d\equiv0,2\pmod6$, $4\nmid d$, $9\nmid d$, and no odd prime congruent to $2$ modulo $3$ divides $d$. Let $\mathcal C_d$ be the Hassett divisor of cubic fourfolds carrying a saturated rank-two algebraic lattice containing the squared hyperplane class and having discriminant $d$. OpenAI claims a threshold $d_0$ beyond which irrationality holds for a very general smooth cubic $X\in\mathcal C_d$ whenever $d$ is admissible but \[ \operatorname{Ku}(X)\simeq D^b(\operatorname{Coh}S) \] for some smooth projective K3 surface $S$, by an exact complex-linear equivalence. Here $\operatorname{Ku}(X)$ is the Kuznetsov component, the orthogonal complement of $\mathcal O_X,\mathcal O_X(1),\mathcal O_X(2)$. Very general excludes countably many proper closed subvarieties. The categorical equivalence is credited to existing K3-category results; irrationality is the new asserted conclusion.

Outline of the argument

Manuscript 1 (family 054, 2026-09-24): An additive invariant tests the fixed integral transcendental K3 lattice through a hypothetical birational map. Sarkisov links express its value as endpoint differences on Mori fiber spaces, using vertical-divisor and generic-surface counts; the resulting telescoping constraint contradicts the large-discriminant lattice data.

Formal verification

Family 054 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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