Congruences of 5secant 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 4folds 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 5secant conics to prove rationality for the first three of the families $C_d$, corresponding to $d=14, 26, 38$ in Hassett's notation.
 Publication:

arXiv eprints
 Pub Date:
 July 2017
 arXiv:
 arXiv:1707.00999
 Bibcode:
 2017arXiv170700999R
 Keywords:

 Mathematics  Algebraic Geometry;
 14E08;
 14J25;
 14N05
 EPrint:
 We 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