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 ${\mathbb{P}}^{55}=\mathbb{P}(H^0({\mathcal{O}}_{{\mathbb{P}}^5}(3)))$ parameterizing all cubic fourfolds 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.