Trisecant flops, their associated K3 surfaces and the rationality of some cubic 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 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 K3 surfaces of degree $d$ contained in well known rational Fano fourfolds.