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Non-positive and negative at infinity divisorial valuations of Hirzebruch surfaces

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Abstract

We consider rational surfaces Z defined by divisorial valuations \(\nu \) of Hirzebruch surfaces. We introduce concepts of non-positivity and negativity at infinity for these valuations and prove that these concepts admit nice local and global equivalent conditions. In particular we prove that, when \(\nu \) is non-positive at infinity, the extremal rays of the cone of curves of Z can be explicitly given.

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Acknowledgements

The authors thank M. Jonsson and W. Veys for valuable comments which helped to improve the paper.

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Authors and Affiliations

  1. Departamento de Matemáticas, Instituto Universitario de Matemáticas y Aplicaciones de Castellón, Jaume I University, Castellón de la Plana, Spain

    Carlos Galindo & Carlos-Jesús Moreno-Ávila

  2. Instituto Universitario de Matemática Pura y Aplicada, Universidad Politécnica de Valencia, Camino de Vera s/n, 46022, Valencia, Spain

    Francisco Monserrat

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  1. Carlos Galindo
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  2. Francisco Monserrat
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Correspondence to Carlos Galindo.

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Partially supported by the Spanish Government Ministerio de Economía, Industria y Competitividad (MINECO), Grants MTM2015-65764-C3-2-P, MTM2016-81735-REDT, PGC2018-096446-B-C22 and BES-2016-076314, as well as by Universitat Jaume I, Grant UJI-B2018-10.

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Galindo, C., Monserrat, F. & Moreno-Ávila, CJ. Non-positive and negative at infinity divisorial valuations of Hirzebruch surfaces. Rev Mat Complut 33, 349–372 (2020). https://doi.org/10.1007/s13163-019-00319-w

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