Computing Cox rings

Hausen, Jürgen; Keicher, Simon; Laface, Antonio

doi:10.1090/mcom/2989

Computing Cox rings
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by Jürgen Hausen, Simon Keicher and Antonio Laface PDF

Math. Comp. 85 (2016), 467-502 Request permission

Abstract:

We consider modifications, for example blow ups, of Mori dream spaces and provide algorithms for investigating the effect on the Cox ring, for example verifying finite generation or computing an explicit presentation in terms of generators and relations. As a first application, we compute the Cox rings of all Gorenstein log del Pezzo surfaces of Picard number one. Moreover, we show computationally that all smooth rational surfaces of Picard numbers at most six are Mori dream surfaces and we provide explicit presentations of the Cox ring for those not admitting a torus action. Finally, we provide the Cox rings of projective spaces blown up at certain special point configurations.

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