# Quotients of toric varieties by actions of subtori

Colloquium Mathematicae (1999)

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

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topŚwięcicka, 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

top- [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] 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] A. Białynicki-Birula and J. Święcicka, A reduction theorem for existence of good quotients, Amer. J. Math. 113 (1991), 189-201. Zbl0741.14031
- [4] A. Białynicki-Birula and J. Święcicka, Three theorems on existence of good quotients, Math. Ann. 307 (1997), 143-149. Zbl0870.14034
- [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] D. A. Cox, The homogeneous coordinate ring of a toric variety, J. Algebraic Geom. 4 (1995), 17-50. Zbl0846.14032
- [7] M. M. Kapranov, B. Sturmfels, and A. V. Zelevinsky, Quotients of toric varieties, Math. Ann. 290 (1991), 643-655. Zbl0762.14023
- [8] D. Mumford, Geometric Invariant Theory, Ergeb. Math. Grenzgeb. 34, Springer, 1982. Zbl0504.14008
- [9] T. Oda, Convex Bodies and Algebraic Geometry, Ergeb. Math. Grenzgeb. 15, Springer, 1988. Zbl0628.52002

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