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

top
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

top

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

top
  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 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.