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

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

Cox rings, total coordinate rings, del Pezzo surfaces
Mathematical Subject Classification 2000
Primary: 14J26
Received: 8 March 2008
Revised: 5 June 2009
Accepted: 14 September 2009
Published: 29 November 2009
Damiano Testa
Mathematical Institute
24-29 St Giles’
Oxford OX1 3LB
United Kingdom
Anthony Várilly-Alvarado
Department of Mathematics
Rice University
MS 136
Houston, TX 77005
United States
Mauricio Velasco
Department of Mathematics
University of California
Berkeley, CA 94720
United States