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

Abstract

We prove the rationality of some Fano fourfolds via Mori Theory and the Minimal Model Program. The method shows a connection between some admissible cubic fourfolds and some birational models of their associated K3 surfaces, pointing out that in these cases rationality may be closely related to the construction of special projections of the associated K3 surfaces. We provide several applications of our method among which we distinguish the solution of the Kuznetsov Conjecture for d = 42 (the first open case) and new proofs for d = 14, 26 and 38. A lot of relevant examples are included, opening the way to further applications of this circle of ideas to the rationality of other classes of Fano fourfolds.

The unreasonable effectiveness of wall-crossing in algebraic geometry

“…We then use wall-crossing for S [5] , interpreted as a moduli space in Ku(Y ) = D b (S), and restrict the resulting birational transformations to Bl Σ Y . In the case a = 2, this recovers completely the picture described in [69] and thus the rationality of all cubics in C 38 . Analogous constructions likely exist for arbitrary d ≡ 2 (mod 6) if we replace Γ s with a locus of sheaves not locally free at s inside a moduli space of stable sheaves.…”

Categorical Torelli theorems: results and open problems

“…None of the two implications is clear but the conjecture perfectly matches the classical Hodge theoretic still conjectural characterization of rational cubic fourfolds due to Harris and Hassett. Some important results in this direction are contained in [133,134].…”

Categorical Torelli theorems: results and open problems

“…None of the two implications is clear but the conjecture perfectly matches the classical Hodge theoretic still conjectural characterization of rational cubic fourfolds due to Harris and Hassett. Some important results in this direction are contained in [118,119].…”