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Forum Mathematicum

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Volume 28, Issue 6 (Nov 2016)

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When are Zariski chambers numerically determined?

Sławomir Rams
  • Institute of Mathematics, Jagiellonian University, ul. Łojasiewicza 6, 30-348 Kraków, Poland; and Institut für Algebraische Geometrie, Leibniz Universität, Welfengarten 1, 30167 Hannover, Germany
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/ Tomasz SzembergORCID iD: http://orcid.org/0000-0002-4234-5838
Published Online: 2016-05-01 | DOI: https://doi.org/10.1515/forum-2015-0087

Abstract

The big cone of every smooth projective surface X admits a natural decomposition into Zariski chambers. The purpose of this note is to give a simple criterion for the interiors of all Zariski chambers on X to be numerically determined Weyl chambers. Such a criterion generalizes the results of Bauer–Funke [4] on K3 surfaces to arbitrary smooth projective surfaces. In the last section, we study the relation between decompositions of the big cone and elliptic fibrations on some surfaces.

Keywords: Zariski decomposition; big cone; elliptic fibration

MSC 2010: 14C20; 14J28

References

  • [1]

    Badescu L., Algebraic Surfaces, Universitext, Springer, New York, 2001. Google Scholar

  • [2]

    Barth W., Hulek K., Peters C. and van de Ven A., Compact Complex Surfaces, 2nd ed., Ergeb. Math. Grenzgeb. (3) 4, Springer, Berlin, 2004. Google Scholar

  • [3]

    Bauer T., A simple proof for the existence of Zariski decompositions on surfaces, J. Algebraic Geom. 18 (2009), no. 4, 789–793. Google Scholar

  • [4]

    Bauer T. and Funke M., Weyl and Zariski chambers on K3 surfaces, Forum Math. 24 (2012), no. 3, 609–625. Google Scholar

  • [5]

    Bauer T., Küronya A. and Szemberg T., Zariski chambers, volumes, and stable base loci, J. Reine Angew. Math. 576 (2004), 209–233. Google Scholar

  • [6]

    Cossec F. R., On the Picard group of Enriques surfaces, Math. Ann. 271 (1985), 577–600. Google Scholar

  • [7]

    Cossec F. R. and Dolgachev I. V., Smooth rational curves on Enriques surfaces, Math. Ann. 272 (1985), 369–384. Google Scholar

  • [8]

    Cossec F. R. and Dolgachev I. V., Enriques Surfaces. I, Progr. Math. 76, Birkhäuser, Basel, 1989. Google Scholar

  • [9]

    Dolgachev I. V., A brief introduction to Enriques surfaces, Development of Moduli Theory (Kyoto 2013), Adv. Stud. Pure Math. 69, Mathematical Society of Japan, Tokyo. (2016), to appear. Google Scholar

  • [10]

    Lazarsfeld R., Positivity in Algebraic Geometry I, II, Springer, Berlin, 2004. Google Scholar

  • [11]

    Nikulin V. V., Description of automorphism groups of Enriques surfaces, Dokl. Akad. Nauk SSSR 277 (1984), no. 6, 1324–1327. Google Scholar

  • [12]

    Rams S. and Schütt M., On Enriques surfaces with four cusps, preprint 2014, http://arxiv.org/abs/1404.3924.

  • [13]

    Schütt M. and Shioda T., Elliptic surfaces, Algebraic Geometry in East Asia (Seoul 2008), Adv. Stud. Pure Math. 60, Mathematical Society of Japan, Tokyo (2010), 51–160. Google Scholar

  • [14]

    Schütt M., Shioda T. and van Luijk R., Lines on Fermat surfaces, J. Number Theory 130 (2010), no. 9, 1939–1963. Google Scholar

  • [15]

    Serre J.-P., A Course in Arithmetic, Grad. Texts in Math. 7, Springer, New York, 1973. Google Scholar

  • [16]

    Zariski O., The theorem of Riemann–Roch for high multiples of an effective divisor on an algebraic surface, Ann. of Math. (2) 76 (1962), 560–615. Google Scholar

About the article


Received: 2015-05-12

Revised: 2015-11-28

Published Online: 2016-05-01

Published in Print: 2016-11-01


Funding Source: Narodowe Centrum Nauki

Award identifier / Grant number: N N201 608040

Award identifier / Grant number: 2014/15/B/ST1/02197

The work was partially supported by National Science Centre, Poland, grant no. N N201 608040 (first author) and grant no. 2014/15/B/ST1/02197 (second author).


Citation Information: Forum Mathematicum, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: https://doi.org/10.1515/forum-2015-0087.

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