Cox Rings of Degree One Del Pezzo Surfaces
نویسندگان
چکیده
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.
منابع مشابه
ar X iv : 0 90 1 . 03 69 v 3 [ m at h . A G ] 3 S ep 2 00 9 ON COX RINGS OF K 3 - SURFACES
We study Cox rings of K3-surfaces. A first result is that a K3surface has a finitely generated Cox ring if and only if its effective cone is rational polyhedral. Moreover, we investigate degrees of generators and relations for Cox rings of K3-surfaces of Picard number two, and explicitly compute the Cox rings of generic K3-surfaces with a non-symplectic involution that have Picard number 2 to 5...
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