Cox rings of degree one del Pezzo surfaces

TL;DR

This paper proves that the Cox ring of a degree one del Pezzo surface is quadratic, confirming a conjecture by Batyrev and Popov, using homological algebra techniques to analyze its minimal free resolution.

Contribution

It completes the proof of the conjecture that Cox rings of degree one del Pezzo surfaces are quadratic algebras, filling a previously unresolved case.

Findings

Cox(X) is a quadratic algebra for degree one del Pezzo surfaces

Homology of a vector space complex determines the minimal free resolution

Many Betti numbers vanish, confirming the conjecture

Abstract

Let X be a del Pezzo surface of degree one over an algebraically closed field (of any characteristic), 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.