Quotients of an affine variety by an action of a torus

Olga Chuvashova; Nikolay Pechenkin

Open Mathematics (2013)

  • Volume: 11, Issue: 11, page 1863-1880
  • ISSN: 2391-5455

Abstract

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Let X be an affine T-variety. We study two different quotients for the action of T on X: the toric Chow quotient X/C T and the toric Hilbert scheme H. We introduce a notion of the main component H 0 of H, which parameterizes general T-orbit closures in X and their flat limits. The main component U 0 of the universal family U over H is a preimage of H 0. We define an analogue of a universal family WX over the main component of X/C T. We show that the toric Chow morphism restricted on the main components lifts to a birational projective morphism from U 0 to W X. The variety W X also provides a geometric realization of the Altmann-Hausen family. In particular, the notion of W X allows us to provide an explicit description of the fan of the Altmann-Hausen family in the toric case.

How to cite

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Olga Chuvashova, and Nikolay Pechenkin. "Quotients of an affine variety by an action of a torus." Open Mathematics 11.11 (2013): 1863-1880. <http://eudml.org/doc/269742>.

@article{OlgaChuvashova2013,
abstract = {Let X be an affine T-variety. We study two different quotients for the action of T on X: the toric Chow quotient X/C T and the toric Hilbert scheme H. We introduce a notion of the main component H 0 of H, which parameterizes general T-orbit closures in X and their flat limits. The main component U 0 of the universal family U over H is a preimage of H 0. We define an analogue of a universal family WX over the main component of X/C T. We show that the toric Chow morphism restricted on the main components lifts to a birational projective morphism from U 0 to W X. The variety W X also provides a geometric realization of the Altmann-Hausen family. In particular, the notion of W X allows us to provide an explicit description of the fan of the Altmann-Hausen family in the toric case.},
author = {Olga Chuvashova, Nikolay Pechenkin},
journal = {Open Mathematics},
keywords = {Torus action; Toric variety; Toric Chow quotient; Toric Hilbert scheme; torus action; toric variety; toric Chow quotient; toric Hilbert scheme},
language = {eng},
number = {11},
pages = {1863-1880},
title = {Quotients of an affine variety by an action of a torus},
url = {http://eudml.org/doc/269742},
volume = {11},
year = {2013},
}

TY - JOUR
AU - Olga Chuvashova
AU - Nikolay Pechenkin
TI - Quotients of an affine variety by an action of a torus
JO - Open Mathematics
PY - 2013
VL - 11
IS - 11
SP - 1863
EP - 1880
AB - Let X be an affine T-variety. We study two different quotients for the action of T on X: the toric Chow quotient X/C T and the toric Hilbert scheme H. We introduce a notion of the main component H 0 of H, which parameterizes general T-orbit closures in X and their flat limits. The main component U 0 of the universal family U over H is a preimage of H 0. We define an analogue of a universal family WX over the main component of X/C T. We show that the toric Chow morphism restricted on the main components lifts to a birational projective morphism from U 0 to W X. The variety W X also provides a geometric realization of the Altmann-Hausen family. In particular, the notion of W X allows us to provide an explicit description of the fan of the Altmann-Hausen family in the toric case.
LA - eng
KW - Torus action; Toric variety; Toric Chow quotient; Toric Hilbert scheme; torus action; toric variety; toric Chow quotient; toric Hilbert scheme
UR - http://eudml.org/doc/269742
ER -

References

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  1. [1] Alexeev V., Brion M., Moduli of affine schemes with reductive group action, J. Algebraic Geom., 2005, 14(1), 83–117 http://dx.doi.org/10.1090/S1056-3911-04-00377-7 Zbl1081.14005
  2. [2] Altmann K., Hausen J., Polyhedral divisors and algebraic torus actions, Math. Ann., 2006, 334(3), 557–607 http://dx.doi.org/10.1007/s00208-005-0705-8 Zbl1193.14060
  3. [3] Arzhantsev I.V., Hausen J., On the multiplication map of a multigraded algebra, Math. Res. Lett., 2007, 14(1), 129–136 Zbl1125.13001
  4. [4] Berchtold F., Hausen J., GIT-equivalence beyond the ample cone, Michigan Math. J., 2006, 54(3), 483–515 http://dx.doi.org/10.1307/mmj/1163789912 Zbl1171.14028
  5. [5] Bertin J., The punctual Hilbert scheme: an introduction, available at http://cel.archives-ouvertes.fr/cel-00437713/en/ Zbl1327.13026
  6. [6] Brion M., Invariant Hilbert schemes, preprint available at http://arxiv.org/abs/1102.0198 
  7. [7] Chuvashova O.V., The main component of the toric Hilbert scheme, Tôhoku Math. J., 2008, 60(3), 365–382 http://dx.doi.org/10.2748/tmj/1223057734 Zbl1160.14001
  8. [8] Cox D.A., Little J.B., Schenck H.K., Toric Varieties, Grad. Stud. Math., 124, American Mathematical Society, Providence, 2011 
  9. [9] Craw A., Maclagan D., Fiber fans and toric quotients, Discrete Comput. Geom., 2007, 37(2), 251–266 http://dx.doi.org/10.1007/s00454-006-1282-7 Zbl1147.14028
  10. [10] Eisenbud D., Harris J., The Geometry of Schemes, Grad. Texts in Math., 197, Springer, New York, 2000 
  11. [11] Fulton W., Introduction to Toric Varieties, Ann. of Math. Stud., 131, Princeton University Press, Princeton, 1993 Zbl0813.14039
  12. [12] Grothendieck A., Éléments de Géométrie Algébrique IV. Étude Locale des Schémas et des Morphismes de Schémas IV, Inst. Hautes Études Sci. Publ. Math., 32, Paris, 1967 Zbl0153.22301
  13. [13] Haiman M., Sturmfels B., Multigraded Hilbert schemes, J. Algebraic Geom., 2004, 13(4), 725–769 http://dx.doi.org/10.1090/S1056-3911-04-00373-X 
  14. [14] Hartshorne R., Algebraic Geometry, Grad. Texts in Math., 52, Springer, New York-Heidelberg, 1977 http://dx.doi.org/10.1007/978-1-4757-3849-0 
  15. [15] Kapranov M.M., Sturmfels B., Zelevinsky A.V., Quotients of toric varieties, Math. Ann., 1991, 290(4), 644–655 Zbl0762.14023
  16. [16] Mumford D., Fogarty J., Kirwan F., Geometric Invariant Theory, 3rd ed., Ergeb. Math. Grenzgeb., 34, Springer, Berlin, 1994 http://dx.doi.org/10.1007/978-3-642-57916-5 Zbl0797.14004
  17. [17] Oda T., Convex Bodies and Algebraic Geometry, Ergeb. Math. Grenzgeb., 15, Springer, Berlin, 1988 
  18. [18] Peeva I., Stillman M., Toric Hilbert schemes, Duke Math. J., 2002, 111(3), 419–449 http://dx.doi.org/10.1215/S0012-7094-02-11132-6 Zbl1067.14005
  19. [19] Swiecicka J., Quotients of toric varieties by actions of subtori, Colloq. Math., 1999, 82(1), 105–116 Zbl0961.14032
  20. [20] Vollmert R., Toroidal embeddings and polyhedral divisors, Int. J. Algebra, 2010, 4(5–8), 383–388 Zbl1210.14058
  21. [21] Ziegler G., Lectures on Polytopes, Grad. Texts in Math., 152, Springer, New York, 1995 http://dx.doi.org/10.1007/978-1-4613-8431-1 

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