Browse › Algebraic and complex geometry › 054 Irrational cubic fourfolds with Hodge-theoretic and categorical K3 associations
Irrational cubic fourfolds with geometric K3 categories
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Abstract
For every sufficiently large admissible Hassett discriminant, we prove that a very general cubic fourfold of that discriminant is irrational, although its Kuznetsov component is equivalent to the ordinary derived category of a projective K3 surface. The discriminant threshold is ineffective. This disproves Kuznetsov's rationality conjecture. The same cubics have associated untwisted polarized K3 surfaces in the Hodge-theoretic sense, so they also disprove the sufficiency direction of the associated-K3 rationality prediction.
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Possibly relevant literature (found by the agent, not checked)
- Bolognesi, Russo, Staglianò, Some loci of rational cubic fourfolds (arXiv:1504.05863, Math. Ann.). Proves that every cubic in the divisor C_14 is rational; this is a further instance of the paper's remark that some Hassett divisors consist entirely of rational cubics, where the paper cites only C_26, C_38 and C_42.
- D. Huybrechts, The K3 category of a cubic fourfold (arXiv:1505.01775, Compositio Math. 153 (2017)). Extends the Addington–Thomas comparison of Hodge-theoretic and categorical K3 associations to twisted K3 surfaces; this bears on the untwisted categorical and Hodge-theoretic hypotheses in Theorem 1.1 and Corollary 1.2.
- Luiz Carazolli, Leonardo F. Cavenaghi, Ludmil Katzarkov, Maxim Kontsevich and Pedro Antonio Muniz Martins, A quantum connection approach to rationality (arXiv:2610.06669v1, 5 October 2026). Its preservation of ranks and regularized exponents of selected quantum-connection factors through blowup comparisons offers a methodological comparison for the manuscript's claimed preservation of whole integral transcendental surface contributions under weak factorization.
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 054
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Irrational-cubic-fourfolds-with-geometric-K3-categories-September-24-2026,
author = {{OpenAI}},
title = {{Irrational cubic fourfolds with geometric K3 categories}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Irrational-cubic-fourfolds-with-geometric-K3-categories-September-24-2026/Irrational-cubic-fourfolds-with-geometric-K3-categories-September-24-2026.pdf}{OAI:Irrational-cubic-fourfolds-with-geometric-K3-categories-September-24-2026}},
year = {2026}
}Family 054: Irrational cubic fourfolds with Hodge-theoretic and categorical K3 associations
For every sufficiently large admissible Hassett discriminant, a very general smooth complex cubic fourfold is irrational despite having both an untwisted geometric K3 category and an integral Hodge-theoretic K3 association. This disproves Kuznetsov's rationality conjecture and the sufficiency of the associated-K3 criterion for rationality; the discriminant threshold is ineffective.
This is the only paper in this family.