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4:00 pm Tuesday, October 4, 2011 Algebraic Geometry Seminar: Rationality of families of intersections of quadrics and derived categoriesby
Asher Auel (Emory) in HB 227- Inspired by the Hodge theoretic study of cubic hypersurfaces and their moduli, Kuznetsov has conjectured a derived categorical condition for rationality of a smooth cubic fourfold: that a certain canonical subcategory of its derived category (of coherent sheaves) is equivalent to the derived category of some K3 surface. If the cubic fourfold contains a plane, this canonical subcategory can be completely described in terms of the associated quadric surface bundle and its Clifford invariant. When the Clifford invariant is trivial, Kuznetsov's conjecture can be verified using results of Hassett. We describe an analogue of this conjecture for another class of fourfolds: families of intersections of two four-dimensional quadrics over the projective line. We use the notion of homological projective duality to describe the analogous subcategory in terms of the Clifford invariant of the associated pencil of quadrics. Using results from the theory of quadratic forms over schemes, we also verify the conjecture when the Clifford invariant is trivial. If there's time, I'll discuss how our techniques yield a derived categorical criterion for the rationality of quartic del Pezzo fibrations over the projective line. This is joint work with Marcello Bernardara and Michele Bolognesi.
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