Families of Segre fourfolds with a view to del Pezzo fibrations

Marcello Bernardara (Universite' de Toulouse)

 Fibrations in del Pezzo surfaces of degree 6 are an interesting case of Mori fiber spaces: for example, special cubic fourfold of discriminant 18 admit such a fibration and their rationality is related to it, as shown by Addington-Hassett-Tschinkel-Varilly-Alvarado. Recently, Kuznetsov described a semiorthogonal decomposition for such fibrations. In this talk, I will present a general construction of a Segre fourfold fibration X-->M with simple degenerations. Namely, a flat map X-->M whose general fiber is isomorphic to P^2 \times P^2 with a natural embedding in a P^8-bundle over M. Such an X is described by a double cover S \to M ramified along the degeneracy locus and an Azumaya algebra B of order 3 over S, and comes with a natural Lefschetz decomposition with respect to map into the P^8 bundle. Such a fibration comes with a natural dual fibration Z-->M in determinantal cubic hypersurfaces of P^8 and a (categorical) resolution of such. As an application of this construction, we aim to give a recipe to construct del Pezzo fibrations of degree 6 over M as double linear sections of such an X, and reconstruct Kuznetsov's semiorthogonal decomposition via relative homological projective duality, as well as fibrations in cubic surfaces over M with determinantal generic fiber. This is a work in progress with Addington, Auel and Faenzi.

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