Cox rings of degree one del Pezzo surfaces

Damiano Testa; Anthony Várilly-Alvarado; Mauricio Velasco

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arXiv · 0803.0353

Cox rings of degree one del Pezzo surfaces

Damiano Testa·Anthony Várilly-Alvarado·Mauricio Velasco

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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.

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Damiano Testa, Anthony Várilly-Alvarado, Mauricio Velasco. 2009-09-17. Cox rings of degree one del Pezzo surfaces. https://arxiv.org/abs/0803.0353

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