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Math-2263 · mathematics · Algebraic geometry

Rationally connected irrational varieties with property (*)

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Statement

Definitions and mathematical statement

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$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
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Is there a rationally connected irrational variety $X$ with property (*)?

Recall Property (*) from a previous question was defined as follows. \[ D(X) = \langle \mathscr A_1, \ldots, \mathscr A_m \rangle \] has a semiorthogonal decomposition with $\mathscr A_i \hookrightarrow D(Y_i)$ such that ${\rm dim} Y_i \leq {\rm dim}X-2.$

Formulation references: [S1], [S2].

Short English statement

Is there a rationally connected irrational variety $X$ with property (*)?

Recall Property (*) from a previous question was defined as follows. \[ D(X) = \langle \mathscr A_1, \ldots, \mathscr A_m \rangle \] has a semiorthogonal decomposition with $\mathscr A_i \hookrightarrow D(Y_i)$ such that ${\rm dim} Y_i \leq {\rm dim}X-2.$

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Statement licence: CC-BY-4.0 · Source file

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Math-2263-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2263

The manuscript claims irrationality for very general cubic fourfolds at sufficiently large admissible discriminants, together with an equivalence of their Kuznetsov component to the derived category of a K3 surface. A smooth cubic fourfold is Fano and therefore rationally connected. Its semiorthogonal decomposition has that K3 component and three exceptional line bundles; these pieces come from varieties of dimensions two and zero, both at most $4-2$. This gives the example requested by the recorded question. The conclusion here is irrationality; stable irrationality is not asserted.

Rationally connected irrational varieties with property $*$

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.

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