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Cox rings of rational surfaces and redundant blow-ups


Authors: DongSeon Hwang and Jinhyung Park
Journal: Trans. Amer. Math. Soc. 368 (2016), 7727-7743
MSC (2010): Primary 14J26; Secondary 14C20
Published electronically: December 18, 2015
MathSciNet review: 3546781
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Abstract: We prove that the redundant blow-up preserves the finite generation of the Cox ring of a rational surface under a suitable assumption, and we study the birational structure of Mori dream rational surfaces via redundant blow-ups. It turns out that the redundant blow-up completely characterizes birational morphisms of Mori dream rational surfaces with anticanonical Iitaka dimension 0. As an application, we construct new Mori dream rational surfaces with anticanonical Iitaka dimension 0 and $ -\infty $ of arbitrarily large Picard number.


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Additional Information

DongSeon Hwang
Affiliation: Department of Mathematics, Ajou University, Suwon, Korea
Email: dshwang@ajou.ac.kr

Jinhyung Park
Affiliation: Department of Mathematical Sciences, KAIST, Daejeon, Korea
Address at time of publication: School of Mathematics, Korea Institute for Advanced Study, Seoul, Korea
Email: parkjh13@kaist.ac.kr

DOI: https://doi.org/10.1090/tran/6611
Keywords: Redundant blow-up, rational surface, Cox ring, Mori dream space, Zariski decomposition
Received by editor(s): May 6, 2014
Received by editor(s) in revised form: November 3, 2014
Published electronically: December 18, 2015
Additional Notes: The first author was partially supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2011-0022904). The second author was partially supported by TJ Park Science Fellowship for Ph.D. Students.
Article copyright: © Copyright 2015 American Mathematical Society