#### 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