Quotients of toric varieties by actions of subtori

Joanna Święcicka

Colloquium Mathematicae (1999)

  • Volume: 82, Issue: 1, page 105-116
  • ISSN: 0010-1354

Abstract

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Let X be an algebraic toric variety with respect to an action of an algebraic torus S. Let Σ be the corresponding fan. The aim of this paper is to investigate open subsets of X with a good quotient by the (induced) action of a subtorus T ⊂ S. It turns out that it is enough to consider open S-invariant subsets of X with a good quotient by T. These subsets can be described by subfans of Σ. We give a description of such subfans and also a description of fans corresponding to quotient varieties. Moreover, we give conditions for a subfan to define an open subset with a complete quotient space.

How to cite

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Święcicka, Joanna. "Quotients of toric varieties by actions of subtori." Colloquium Mathematicae 82.1 (1999): 105-116. <http://eudml.org/doc/210743>.

@article{Święcicka1999,
abstract = {Let X be an algebraic toric variety with respect to an action of an algebraic torus S. Let Σ be the corresponding fan. The aim of this paper is to investigate open subsets of X with a good quotient by the (induced) action of a subtorus T ⊂ S. It turns out that it is enough to consider open S-invariant subsets of X with a good quotient by T. These subsets can be described by subfans of Σ. We give a description of such subfans and also a description of fans corresponding to quotient varieties. Moreover, we give conditions for a subfan to define an open subset with a complete quotient space.},
author = {Święcicka, Joanna},
journal = {Colloquium Mathematicae},
keywords = {group actions; quotients; orbit spaces; quotients of toric varieties; actions of subtori; subfans; complete quotient space},
language = {eng},
number = {1},
pages = {105-116},
title = {Quotients of toric varieties by actions of subtori},
url = {http://eudml.org/doc/210743},
volume = {82},
year = {1999},
}

TY - JOUR
AU - Święcicka, Joanna
TI - Quotients of toric varieties by actions of subtori
JO - Colloquium Mathematicae
PY - 1999
VL - 82
IS - 1
SP - 105
EP - 116
AB - Let X be an algebraic toric variety with respect to an action of an algebraic torus S. Let Σ be the corresponding fan. The aim of this paper is to investigate open subsets of X with a good quotient by the (induced) action of a subtorus T ⊂ S. It turns out that it is enough to consider open S-invariant subsets of X with a good quotient by T. These subsets can be described by subfans of Σ. We give a description of such subfans and also a description of fans corresponding to quotient varieties. Moreover, we give conditions for a subfan to define an open subset with a complete quotient space.
LA - eng
KW - group actions; quotients; orbit spaces; quotients of toric varieties; actions of subtori; subfans; complete quotient space
UR - http://eudml.org/doc/210743
ER -

References

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  1. [1] A. Białynicki-Birula, Finiteness of the number of maximal open sets with a good quotient, Transformation Groups 3 (1998), 301-319. Zbl0940.14034
  2. [2] A. Białynicki-Birula and J. Święcicka, Open subsets of projective spaces with a good quotient by an action of a reductive group, ibid. 1 (1996), 153-185. Zbl0912.14016
  3. [3] A. Białynicki-Birula and J. Święcicka, A reduction theorem for existence of good quotients, Amer. J. Math. 113 (1991), 189-201. Zbl0741.14031
  4. [4] A. Białynicki-Birula and J. Święcicka, Three theorems on existence of good quotients, Math. Ann. 307 (1997), 143-149. Zbl0870.14034
  5. [5] A. Białynicki-Birula and J. Święcicka, A recipe for finding open subsets of vector spaces with good quotient, Colloq. Math. 77 (1998), 97-113. Zbl0947.14027
  6. [6] D. A. Cox, The homogeneous coordinate ring of a toric variety, J. Algebraic Geom. 4 (1995), 17-50. Zbl0846.14032
  7. [7] M. M. Kapranov, B. Sturmfels, and A. V. Zelevinsky, Quotients of toric varieties, Math. Ann. 290 (1991), 643-655. Zbl0762.14023
  8. [8] D. Mumford, Geometric Invariant Theory, Ergeb. Math. Grenzgeb. 34, Springer, 1982. Zbl0504.14008
  9. [9] T. Oda, Convex Bodies and Algebraic Geometry, Ergeb. Math. Grenzgeb. 15, Springer, 1988. Zbl0628.52002

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