Trisecant Flops, their associated K3 surfaces and the rationality of some Fano fourfolds

Abstract

We provide a new construction of rationality for cubic fourfolds via Mori's theory and the minimal model program. As an application, we present the solution of the Kuznetsov's conjecture for $d=42$ (the first open case). Our methods also show an explicit connection between the rationality of cubic fourfolds belonging to the first four admissible families ${\mathcal{C}}_d$, with $d=14,26,38$, and $42$ and some birational models of minimal K3 surfaces of degree $d$ contained in well known rational Fano fourfolds.

New section on the explicit detection of the associated K3 surface. To appear in the Journal of the European Mathematical Society

References in corpus (1)

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

Cited by in corpus (3)

• Rational cubic fourfolds with associated singular K3 surfaces
• New rational cubic fourfolds arising from Cremona transformations
• On lattice polarizable cubic fourfolds