Congruences of 5-secant conics and the rationality of some admissible cubic fourfolds

Abstract

The works of Hassett and Kuznetsov identify countably many divisors $C_d$ in the open subset of ${\mathbb{P}}^{55}=\mathbb{P}(H^0({\mathcal{O}}_{{\mathbb{P}}^5}(3)))$ parametrizing all cubic 4-folds and conjecture that the cubics corresponding to these divisors are precisely the rational ones. Rationality has been known classically for the first family $C_{14}$. We use congruences of 5-secant conics to prove rationality for the first three of the families $C_d$, corresponding to $d=14,26,38$ in Hassett's notation.

arXiv commentWe added more details, improving the presentation, and modified some discursive parts. Theorem 1 has been restated in a weaker form with a hypothesis always satisfied in our applications

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• Rational cubic fourfolds in Hassett divisors
• Rational cubic fourfolds with associated singular K3 surfaces
• New rational cubic fourfolds arising from Cremona transformations
• Hilbert schemes of two points on K3 surfaces and certain rational cubic fourfolds
• Some Moduli of $n$-pointed Fano Fourfolds
• On lattice polarizable cubic fourfolds