## Algebra & Number Theory

### Cox rings of degree one del Pezzo surfaces

#### 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.

#### Article information

Source
Algebra Number Theory, Volume 3, Number 7 (2009), 729-761.

Dates
Received: 8 March 2008
Revised: 5 June 2009
Accepted: 14 September 2009
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.ant/1513797480

Digital Object Identifier
doi:10.2140/ant.2009.3.729

Mathematical Reviews number (MathSciNet)
MR2579393

Zentralblatt MATH identifier
1191.14047

Subjects
Primary: 14J26: Rational and ruled surfaces

#### Citation

Testa, Damiano; Várilly-Alvarado, Anthony; Velasco, Mauricio. Cox rings of degree one del Pezzo surfaces. Algebra Number Theory 3 (2009), no. 7, 729--761. doi:10.2140/ant.2009.3.729. https://projecteuclid.org/euclid.ant/1513797480

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