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

TL;DR

This paper introduces a new approach to establish the rationality of certain cubic fourfolds using Mori's theory, solving a key conjecture and linking rationality to minimal K3 surfaces within Fano fourfolds.

Contribution

It provides a novel construction method for rationality of cubic fourfolds and explicitly connects their rationality to minimal K3 surfaces and known Fano fourfolds.

Findings

Solved Kuznetsov's conjecture for d=42

Connected rationality of cubic fourfolds to minimal K3 surfaces

Developed a new construction method using Mori's theory

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.