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 P 55 = P(H 0 (O 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 C14. 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.

New examples of rational Gushel-Mukai fourfolds

“…This condition can be expressed by saying that the rational cubic fourfolds are parametrized by a countable union d C d of irreducible hypersurfaces inside the 20-dimensional coarse moduli space of cubic fourfolds, where d runs over the so-called admissible values (the first ones are d = 14, 26, 38, 42, 62). The rationality for the cubic fourfolds in C 14 has been classically proved by Fano in [Fan43] (see also [BRS19]), while in [RS19a] it has been proved the rationality in the case of C 26 and C 38 . Very recently, in [RS19b] it has been also proved the rationality in the case of C 42 .…”

Algebraic hypersurfaces

“…The strongest results, due to Russo and Staglianò [RS17], prove rationality for cubic 4-folds corresponding to three of the hypersurfaces H i (i = 14, 26, 38 in Hassett's notation). Just to show that there are some rather subtle rationality constructions, here is one of the beautiful examples they discovered for H 38 .…”

Cubic fourfolds fibered in sextic del Pezzo surfaces

“…Kuznetsov [Kuz17] has further developed the theory of sextic del Pezzo surfaces and their degenerations, with a view toward applications to derived categories. Russo and Staglianò [RS17] have proposed new constructions for rational cubic fourfolds applicable over two divisors in the moduli space.…”