# Mathematics Colloquia and Seminars

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### Picard graded Betti numbers and the Cox rings of degree 1 Del Pezzo surfaces

**Algebra & Discrete Mathematics**

Speaker: | Mauricio Velasco, UC Berkeley |

Location: | 2112 MSB |

Start time: | Fri, Apr 18 2008, 1:10PM |

The 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 fundamental 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 prove a conjecture of Batyrev and Popov stating that the Cox rings of all Del Pezzo surfaces are quadratic algebras.

This is joint work with A. Laface, D. Testa and T. Varilly.

Lunch with speaker at 11:30.