Singular del Pezzo surfaces whose universal torsors are hypersurfaces

We classify all generalized del Pezzo surfaces (that is, minimal desingularizations of singular del Pezzo surfaces containing only rational double points) whose universal torsors are open subsets of hypersurfaces in affine space. Equivalently, their Cox rings are polynomial rings with exactly one re...

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Bibliographic Details
Published in:Proceedings of the London Mathematical Society 2014-03, Vol.108 (3), p.638-681
Main Author: Derenthal, Ulrich
Format: Article
Language:English
Online Access:Get full text
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Summary:We classify all generalized del Pezzo surfaces (that is, minimal desingularizations of singular del Pezzo surfaces containing only rational double points) whose universal torsors are open subsets of hypersurfaces in affine space. Equivalently, their Cox rings are polynomial rings with exactly one relation. For all 30 types with this property, we describe the Cox rings in detail. These explicit descriptions can be applied to study Manin's conjecture on the asymptotic behavior of the number of rational points of bounded height for singular del Pezzo surfaces, using the universal torsor method.
ISSN:0024-6115
1460-244X
DOI:10.1112/plms/pdt041