Let be a del Pezzo surface of degree one over an algebraically closed field, and let be its total coordinate ring. We prove the missing case of a conjecture of Batyrev and Popov, which states that is a quadratic algebra. We use a complex of vector spaces whose homology determines part of the structure of the minimal free -graded resolution of over a polynomial ring. We show that sufficiently many Betti numbers of this minimal free resolution vanish to establish the conjecture.
"Cox rings of degree one del Pezzo surfaces." Algebra Number Theory 3 (7) 729 - 761, 2009. https://doi.org/10.2140/ant.2009.3.729