Seminario de Geometría Algebraica

Barbara Bolognese, Università Roma Tre.

Resumen: A powerful tool of investigation of Fano varieties is provided by exceptional collections in their derived categories. In general, proving the fullness of such a collection is a hard problem, often done on a case-by-case basis, with the aid of a deep understanding of the underlying geometry. Likewise, when an exceptional collection is not full, it is not straightforward to determine whether its residual category is the derived category of a variety.

Taking after Bondal and Orlov, we examine two cases: the case of quadric hypersurfaces in Pn+1 and the case of the index 2 Fano threefold Y (the generic intersection of two quadrics in P5. In the first case, we prove that the classical result by Kapranov on the fullness of the standard exceptions is equivalent to the existence of a numerical stability condition on the residual category of the exceptional collection of the quadric. In the second case, we show how the same technique recovers the equivalence of the residual category of the exceptional collection {OY,OY(1)} with the derived category of a genus 2 curve. This is joint work with Domenico Fiorenza.

https://www.matem.unam.mx/~lozano/eseminar.html