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.
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