Oberseminar SoSe 2012 Zusammenfassung des Vortrags von Michele Bolognesi (Rennes, FRA)
Rational cubic fourfolds containing a plane with nontrivial Clifford invariant
In this talk I will showcase a general class of smooth rational cubic fourfolds X containing a plane whose associated quadric surface bundle does not have a rational section. Equivalently, the Brauer class B of the even Clifford algebra over the discriminant cover (a K3 surface S of degree 2) associated to the quadric bundle, is nontrivial. These fourfolds provide nontrivial examples verifying Kuznetsov's conjecture on the rationality of cubic fourfolds containing a plane. Indeed, using homological projective duality for grassmannians, one obtains another K3 surface S' of degree 14 and a nontrivial twisted derived equivalence AX = D(S;B )= D(S'), where AX is Kuznetsov's residual category associated to the cubic hypersurface X.
This is a joint project with A. Auel, M. Bernardara and A. Varilly Alvarado.