On the Cox ring of Del Pezzo surfaces

2006/03/04 by Ulrich Derenthal, Derenthal, Ulrich · 1 citation
Mathematics · #14J26 (Primary) 14C20 (Secondary) #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.math/0603111

openalex publication_date 2006/03/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let S_r be the blow-up of P² in r general points, i.e., a smooth Del Pezzo surface of degree 9-r. For r <= 7, we determine the quadratic equations defining its Cox ring explicitly. The ideal of the relations in Cox(S₈) is calculated up to radical. As conjectured by Batyrev and Popov, all the generating relations are quadratic.

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• Universal torsors of Del Pezzo surfaces and homogeneous spaces

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