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Mathematics of Computation

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Computing Cox rings


Authors: Jürgen Hausen, Simon Keicher and Antonio Laface
Journal: Math. Comp. 85 (2016), 467-502
MSC (2010): Primary 14L24, 14L30, 14C20, 14Q10, 14Q15, 13A30, 52B55
DOI: https://doi.org/10.1090/mcom/2989
Published electronically: June 10, 2015
MathSciNet review: 3404458
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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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Additional Information

Jürgen Hausen
Affiliation: Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany
Email: juergen.hausen@uni-tuebingen.de

Simon Keicher
Affiliation: Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany
Email: keicher@mail.mathematik.uni-tuebingen.de

Antonio Laface
Affiliation: Departamento de Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile
Email: alaface@udec.cl

DOI: https://doi.org/10.1090/mcom/2989
Received by editor(s): October 20, 3013
Received by editor(s) in revised form: April 24, 2014, July 21, 2014, and August 10, 2014
Published electronically: June 10, 2015
Additional Notes: The second author was partially supported by the DFG Priority Program SPP 1489
The third author was partially supported by Proyecto FONDECYT Regular N. 1110096
Article copyright: © Copyright 2015 American Mathematical Society

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