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Orbit spaces of Small Tori

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Abstract

Consider an algebraic torus of small dimension acting on an open subset of ℂn, or more generally on a quasiaffine variety such that a separated orbit space exists. We discuss under which conditions this orbit space is quasiprojective. One of our counterexamples provides a toric variety with enough effective invariant Cartier divisors that is not embeddable into a smooth toric variety.

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References

  1. A. A’Campo-Neuen, J. Hausen: A non-separated version of Kajiwara’s Construction. Konstanzer Schriften in Mathematik und Informatik 103 (1999)

  2. A. A’Campo-Neuen, J. Hausen: Quotients of divisorial toric varieties. Michigan Math. J., Vol. 50, No. 1, 101–123

  3. P. Berthelot, A. Grothendieck, L. Illusie et. al.: Theorie des Intersections et Théorème de Riemann-Roch, SGA 6, Springer LNM 225 (1971)

  4. A. Bialynicki-Birula, J. Świcecicka: On exotic orbit spaces of tori acting on projective varieties. In: Group actions and invariant theory. Proc. Conf. Montreal 1988 CMS Conf. Proc. 10, 25–30 (1989)

    MathSciNet  Google Scholar 

  5. A. Bialynicki-Birula, J. Swicecicka: A recipe for finding open subsets of vector spaces with a good quotient. Colloq. Math., Vol. 77, No. 1, 97–114 (1998)

    MathSciNet  MATH  Google Scholar 

  6. M. Borelli: Divisorial varieties. Pacific J. Math. 13 (1963), 375–388 [ai]7]_D. Cox: The homogeneous coordinate ring of a toric variety. J. Alg. Geom. 4, 17–50 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  7. G. Ewald: Spherical complexes and nonprojective toric varieties. Discrete Comput. Geom. 1, 115–122 (1986).

    Article  MathSciNet  MATH  Google Scholar 

  8. W. Fulton: Introduction to toric varieties. The William H. Roever Lectures in Geometry, Princeton University Press (1993)

  9. B. Grünbaum: Convex Polytopes. Interscience Publishers, New York, 1967

    MATH  Google Scholar 

  10. H. A. Hamm: Very good quotients of toric varieties. In: Bruce, J. W. (ed.) et al. Real and complex singularities. Proceedings of the 5th workshop; São Carlos, Brazil, July 27–31, 1998. Chapman/Hall/CRC Res. Notes Math. 412, 61–75 (2000)

    Google Scholar 

  11. J. Hausen: Equivariant embeddings into smooth toric varieties. Canad. J. Math., Vol. 54, No. 3, 554–570 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  12. T. Kajiwara: The functor of a toric variety with enough invariant effective Cartier divisors. Tôhoku Math. J., 50 (1998), 139–157

    Article  MathSciNet  MATH  Google Scholar 

  13. P. Kleinschmidt: A classification of toric varieties with few generators. Aequationes Math. 35, No.2/3 (1988), 254–266

    Article  MathSciNet  MATH  Google Scholar 

  14. P. Kleinschmidt, B. Sturmfels: Smooth toric varieties with small Picard number are projective. Topology 30, No. 2 (1991), 289–299

    Article  MathSciNet  MATH  Google Scholar 

  15. P. Mani: Spheres with few vertices. J. Comb. Theor. 13 (1972), 346–352

    Article  MathSciNet  MATH  Google Scholar 

  16. D. Mumford, J. Fogarty, F. Kirwan: Geometric Invariant Theory, 3rd enlarged edition, Springer, Berlin, 1994.

    Book  Google Scholar 

  17. T. Oda: Lectures on Torus Embeddings and Applications, Tata Inst. of Fund. Research 58, Springer, Berlin, Heidelberg, New York, 1978

    Google Scholar 

  18. G.C. Shephard: Spherical complexes and radial projections of polytopes. Israel J. Math. 9, 257–262 (1971).

    Article  MathSciNet  MATH  Google Scholar 

  19. H. Sumihiro: Equivariant completion. J. Math. Kyoto Univ. 14, 1–28 (1974)

    MathSciNet  MATH  Google Scholar 

  20. J. Świcecicka: Quotients of Toric Varieties by Actions of Subtori. Colloq. Math., Vol. 82, No. 1, 105–116 (1999)

    MathSciNet  Google Scholar 

  21. J. Włodarczyk: Embeddings in toric varieties and prevarieties. J. Alg. Geometry 2, 705–726 (1993)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Fachbereich Mathematik, Johannes-Gutenberg-Universität Mainz, 55099, Mainz, Germany

    Annette A’campo-Neuen

  2. Fachbereich Mathematik und Statistik, Universität Konstanz, 78457, Konstanz, Germany

    Jürgen Hausen

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  1. Annette A’campo-Neuen
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  2. Jürgen Hausen
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Correspondence to Annette A’campo-Neuen.

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A’campo-Neuen, A., Hausen, J. Orbit spaces of Small Tori. Results. Math. 43, 13–22 (2003). https://doi.org/10.1007/BF03322717

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