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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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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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Additional Information
  • Jürgen Hausen
  • Affiliation: Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany
  • MR Author ID: 361664
  • Email: juergen.hausen@uni-tuebingen.de
  • Simon Keicher
  • Affiliation: Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany
  • MR Author ID: 1001701
  • Email: keicher@mail.mathematik.uni-tuebingen.de
  • Antonio Laface
  • Affiliation: Departamento de Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile
  • MR Author ID: 634848
  • Email: alaface@udec.cl
  • 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
  • © Copyright 2015 American Mathematical Society
  • 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
  • MathSciNet review: 3404458