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Gluing Affine Torus Actions Via Divisorial Fans

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Abstract

Generalizing the passage from a fan to a toric variety, we provide a combinatorial approach to construct arbitrary effective torus actions on normal, algebraic varieties. Based on the notion of a “proper polyhedral divisor” introduced in earlier work, we develop the concept of a “divisorial fan” and show that these objects encode the equivariant gluing of affine varieties with torus action. We characterize separateness and completeness of the resulting varieties in terms of divisorial fans, and we study examples like \( \mathbb{C} \)*-surfaces and projectivizations of (nonsplit) vector bundles over toric varieties.

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References

  1. K. Altmann, J. Hausen, Polyhedral divisors and algebraic torus actions, Math. Ann. 334 (2006), 557–607

    Article  MATH  MathSciNet  Google Scholar 

  2. H. Flenner, S. Kaliman, M. Zaidenberg, Completions of \( \mathbb{C} \)*-surfaces, in: Affine Algebraic Geometry, Osaka University Press, Osaka, 2007, pp. 149–201.

    Google Scholar 

  3. R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics, Vol. 52, Springer-Verlag, New York, 1977. Russian tranl.: P. Xapтсхорн, Aлƨебраuческая ƨеомеmpuя, Mиp, M, 1981.

  4. J. Hausen, Complete orbit spaces of affine torus actions, to appear in Internat. J. Math., Eprint, arXiv: math AG/0510489.

  5. G. Kempf, F. Knudson, D. Mumford, B. Saint-Donat, Toroidal Embeddings I, Lecture Notes in Mathematics, Vol. 339, Springer-Verlag, New York, 1973.

    Google Scholar 

  6. A. A. Клячко, Экеuеарuанmные расслоенuя на mорuческuх мноƨообразuях, Изв, AH CCCP, Cep. мaт. 53 (1989), no. 5, 1001–1039. Engl. transl.: A. A. Klyachko, Equivariant bundles on toral varieties, Math. USSR-Izv. 35 (1990), no. 2, 337–375.

  7. M. Perling, Graded rings and equivariant sheaves on toric varieties, Math. Nachr. 263–64 (2004), 181–197.

    Article  MathSciNet  Google Scholar 

  8. H. Sumihiro, Equivariant completions, J. Math. Kyoto Univ. 14 (1974), 1–28.

    MATH  MathSciNet  Google Scholar 

  9. R. Vollmert, Toroidal embeddings and polyhedral divisors, Eprint, arXiv: math.AG/0707.0917v1.

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Authors and Affiliations

  1. Fachbereich Mathematik und Informatik, Freie Universität Berlin Arnimalle 3, 14195, Berlin, Germany

    Klaus Altmann

  2. Mathematisches Institut, Universität Tübingen Auf der Morgenstelle 10, 72076, Tübingen, Germany

    Jürgen Hausen

  3. Institut für Mathematik LS Algebra und Geometrie Brandenburgische Technische, Universität Cottbus, PF 10 13 44, 03013, Cottbus, Germany

    Hendrik Süss

Authors
  1. Klaus Altmann
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  2. Jürgen Hausen
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  3. Hendrik Süss
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Correspondence to Klaus Altmann.

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Altmann, K., Hausen, J. & Süss, H. Gluing Affine Torus Actions Via Divisorial Fans. Transformation Groups 13, 215–242 (2008). https://doi.org/10.1007/s00031-008-9011-3

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  • DOI: https://doi.org/10.1007/s00031-008-9011-3

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