Francesco Russo

Title: Rationality of cubic fourfolds via Trisecant Flops and via (non minimal) associated K3 surfaces

Abstract: 
According to Kuznetsov Conjecture, in the moduli space of cubic fourfolds there exist infinitely many
irreducible divisors (cubics of "admissible discriminant d" in the sense of Hassett), whose union should 
be  the locus of rational cubic fourfolds. Via the construction of the Trisecant Flops and via the theory of 
the congruences of $3e-1$-secant curves of degree $e$ to surfaces in P5, we shall explain the role played 
by the "associated" (non minimal) K3 surfaces in various rationality questions regarding cubic and
Gushel-Mukai fourfolds. As an application we shall present uniform proofs of the cases d=14, 26, 38 and 42
of the Conjecture, classically known only for d=14 (Fano, 1943), and discuss further possible developments 
of this circle of ideas. This is joint work with Giovanni Stagliano.