Trisecant flops, their associated K3 surfaces and the rationality of some cubic fourfolds

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 ratio-nality of cubic fourfolds belonging to the first four admissible families Cd with d = 14, 26, 38, and 42 and some birational models of K3 surfaces of degree d contained in well known rational Fano fourfolds.