3:45: Picard-graded betti numbers and Cox ringsMauricio VelascoThe Cox ring of an algebraic variety X fits in the following analogy: Cox(X) is to X as the ring of polynomials k[x0,...,xn] is to projective space Pn.It is known that the Cox ring of X is a polynomial ring iff X is Toric and that there is a large class of varieties, the so called Mori Dream Spaces, whose Cox rings are finitely generated algebras, that is, Cox(X)= S/I for a homogeneous ideal I in a Pic(X)-graded polynomial ring S. The question of describing the ideal I and of understanding how it relates with the geometry of the variety is a fundamentally open problem. The purpose of this talk is to introduce a tool to investigate this question. We define complexes of vector spaces whose homology determines the Pic(X)-graded Betti numbers of Cox(X) and we show that these complexes can be studied with the methods of complex algebraic geometry (i.e. via Riemann-Roch and the Kawamata-Viehweg vanishing theorem). As an application of this technique we give a new proof of the fact that the Cox rings of Del Pezzo surfaces (of degree >1) are quadratic algebras. These results are joint work with Antonio Laface and Greg Smith. |
5:00: Matrix factorization in the twenty-first centuryDavid Morrison |