#### Vol. 3, No. 7, 2009

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Cox rings of degree one del Pezzo surfaces

### Damiano Testa, Anthony Várilly-Alvarado and Mauricio Velasco

Vol. 3 (2009), No. 7, 729–761
##### Abstract

Let $X$ be a del Pezzo surface of degree one over an algebraically closed field, and let $Cox\left(X\right)$ be its total coordinate ring. We prove the missing case of a conjecture of Batyrev and Popov, which states that $Cox\left(X\right)$ is a quadratic algebra. We use a complex of vector spaces whose homology determines part of the structure of the minimal free $Pic\left(X\right)$-graded resolution of $Cox\left(X\right)$ over a polynomial ring. We show that sufficiently many Betti numbers of this minimal free resolution vanish to establish the conjecture.

##### Keywords
Cox rings, total coordinate rings, del Pezzo surfaces
Primary: 14J26
##### Milestones
Received: 8 March 2008
Revised: 5 June 2009
Accepted: 14 September 2009
Published: 29 November 2009
##### Authors
 Damiano Testa Mathematical Institute 24-29 St Giles’ Oxford OX1 3LB United Kingdom http://www.maths.ox.ac.uk/node/7850 Anthony Várilly-Alvarado Department of Mathematics Rice University MS 136 Houston, TX 77005 United States http://math.rice.edu/~av15 Mauricio Velasco Department of Mathematics University of California Berkeley, CA 94720 United States http://math.berkeley.edu/~velasco