Cox rings of degree one del Pezzo surfaces

Testa, Damiano; Várilly-Alvarado, Anthony; Velasco, Mauricio

doi:10.2140/ant.2009.3.729

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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

DOI: 10.2140/ant.2009.3.729

Abstract

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.

Keywords

Cox rings, total coordinate rings, del Pezzo surfaces

Mathematical Subject Classification 2000

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

Vol. 3, No. 7, 2009