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Kuznetsov's rationality conjecture for cubic fourfolds

For a smooth complex cubic fourfold X⊂P5X\subset\mathbf P^5, the Kuznetsov component Ku(X)\mathrm{Ku}(X) is the right orthogonal of OX,OX(1),OX(2)\mathcal O_X,\mathcal O_X(1),\mathcal O_X(2) in Db(X)D^b(X), a K3 category. All known rational cubics (Pfaffian, containing a plane with a section, divisors C26,C38,C42\mathcal C_{26},\mathcal C_{38},\mathcal C_{42}) have Ku(X)\mathrm{Ku}(X) equivalent to the derived category of a K3 surface, and by Addington-Thomas this matches Hassett's lattice condition for associated K3 surfaces. Kuznetsov conjectured that XX is rational if and only if Ku(X)≃Db(S)\mathrm{Ku}(X)\simeq D^b(S) for a projective K3 surface SS. Is that true?

Result
Disproved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Algebraic geometry: rationality, derived categories, cubic fourfolds
Posed by
Alexander Kuznetsov (Derived categories of cubic fourfolds, 2010, Conjecture 1.1)
Year posed
2010
Years open
16y
Solved
2026-09-24
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
45 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Claims that for every sufficiently large admissible Hassett discriminant dd a very general cubic fourfold of discriminant dd is irrational although its Kuznetsov component is the derived category of a K3 surface and it has an associated polarized K3 in Hassett's Hodge-theoretic sense. This disproves Kuznetsov's conjecture and the sufficiency of the associated-K3 criterion (and of Fano variety birational to a K3 Hilbert square). The threshold is ineffective; no specific discriminant is shown irrational, the necessity direction is untouched, and stable rationality is not decided. These are not the first irrational cubic fourfolds: the paper cites Katzarkov-Kontsevich-Pantev-Yu (very general cubic) and Fay (some special divisors).

What the AI did

The OpenAI math release (github.com/openai/math) states that its results were produced by an unreleased internal OpenAI model, with on average about three hours of ChatGPT Pro thinking compute per result, under one fixed procedure applied to roughly 4,000 posed problems; outputs were then grouped into families and filtered for significance. The manuscript is credited to OpenAI alone and names no human author.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 and Corollaries 1.2 and 1.3 of the TeX source read against the posed problem. For every admissible dd above an ineffective d0d_0, a very general cubic in Cd\mathcal C_d is irrational yet has Ku(X)≃Db(S)\mathrm{Ku}(X)\simeq D^b(S); this refutes the categorical-implies-rational direction, which is enough to disprove the conjecture. The categorical equivalence comes from published work (Addington-Thomas; Bayer, Lahoz, Macri, Nuer, Perry, Stellari); the new part is irrationality, via an additive invariant of birational maps computed by weak factorization with Gromov-Witten operators and by Sarkisov links. Not refereed. No Lean formalization is listed.

Sources

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