Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

1 Issue per year


IMPACT FACTOR 2016 (Open Mathematics): 0.682
IMPACT FACTOR 2016 (Central European Journal of Mathematics): 0.489

CiteScore 2017: 0.68

SCImago Journal Rank (SJR) 2017: 0.450
Source Normalized Impact per Paper (SNIP) 2017: 0.829

Mathematical Citation Quotient (MCQ) 2016: 0.23

Open Access
Online
ISSN
2391-5455
See all formats and pricing
More options …
Volume 11, Issue 11

Issues

Volume 13 (2015)

Quotients of an affine variety by an action of a torus

Olga Chuvashova
  • Department of Higher Algebra, Faculty of Mechanics and Mathematics, Moscow State University, 1 Leninskie Gory, Moscow, 119991, Russia
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Nikolay Pechenkin
Published Online: 2013-08-23 | DOI: https://doi.org/10.2478/s11533-013-0295-8

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.

MSC: 14L24; 14C05; 14M25

Keywords: Torus action; Toric variety; Toric Chow quotient; Toric Hilbert scheme

  • [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/S1056-3911-04-00377-7CrossrefGoogle Scholar

  • [2] Altmann K., Hausen J., Polyhedral divisors and algebraic torus actions, Math. Ann., 2006, 334(3), 557–607 http://dx.doi.org/10.1007/s00208-005-0705-8CrossrefGoogle Scholar

  • [3] Arzhantsev I.V., Hausen J., On the multiplication map of a multigraded algebra, Math. Res. Lett., 2007, 14(1), 129–136 CrossrefGoogle Scholar

  • [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/1163789912CrossrefGoogle Scholar

  • [5] Bertin J., The punctual Hilbert scheme: an introduction, available at http://cel.archives-ouvertes.fr/cel-00437713/en/ Google Scholar

  • [6] Brion M., Invariant Hilbert schemes, preprint available at http://arxiv.org/abs/1102.0198 Google Scholar

  • [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/1223057734CrossrefGoogle Scholar

  • [8] Cox D.A., Little J.B., Schenck H.K., Toric Varieties, Grad. Stud. Math., 124, American Mathematical Society, Providence, 2011 Google Scholar

  • [9] Craw A., Maclagan D., Fiber fans and toric quotients, Discrete Comput. Geom., 2007, 37(2), 251–266 http://dx.doi.org/10.1007/s00454-006-1282-7CrossrefGoogle Scholar

  • [10] Eisenbud D., Harris J., The Geometry of Schemes, Grad. Texts in Math., 197, Springer, New York, 2000 Google Scholar

  • [11] Fulton W., Introduction to Toric Varieties, Ann. of Math. Stud., 131, Princeton University Press, Princeton, 1993 Google Scholar

  • [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 Google Scholar

  • [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-XCrossrefGoogle Scholar

  • [14] Hartshorne R., Algebraic Geometry, Grad. Texts in Math., 52, Springer, New York-Heidelberg, 1977 http://dx.doi.org/10.1007/978-1-4757-3849-0CrossrefGoogle Scholar

  • [15] Kapranov M.M., Sturmfels B., Zelevinsky A.V., Quotients of toric varieties, Math. Ann., 1991, 290(4), 644–655 Google Scholar

  • [16] Mumford D., Fogarty J., Kirwan F., Geometric Invariant Theory, 3rd ed., Ergeb. Math. Grenzgeb., 34, Springer, Berlin, 1994 http://dx.doi.org/10.1007/978-3-642-57916-5CrossrefGoogle Scholar

  • [17] Oda T., Convex Bodies and Algebraic Geometry, Ergeb. Math. Grenzgeb., 15, Springer, Berlin, 1988 Google Scholar

  • [18] Peeva I., Stillman M., Toric Hilbert schemes, Duke Math. J., 2002, 111(3), 419–449 http://dx.doi.org/10.1215/S0012-7094-02-11132-6CrossrefGoogle Scholar

  • [19] Swiecicka J., Quotients of toric varieties by actions of subtori, Colloq. Math., 1999, 82(1), 105–116 Google Scholar

  • [20] Vollmert R., Toroidal embeddings and polyhedral divisors, Int. J. Algebra, 2010, 4(5–8), 383–388 Google Scholar

  • [21] Ziegler G., Lectures on Polytopes, Grad. Texts in Math., 152, Springer, New York, 1995 http://dx.doi.org/10.1007/978-1-4613-8431-1CrossrefGoogle Scholar

About the article

Published Online: 2013-08-23

Published in Print: 2013-11-01


Citation Information: Open Mathematics, Volume 11, Issue 11, Pages 1863–1880, ISSN (Online) 2391-5455, DOI: https://doi.org/10.2478/s11533-013-0295-8.

Export Citation

© 2013 Versita Warsaw. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

Comments (0)

Please log in or register to comment.
Log in