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Equivariant Embeddings into Smooth Toric Varieties

Published online by Cambridge University Press:  20 November 2018

Jürgen Hausen
Jürgen Hausen*
Affiliation:
Fachbereich Mathematik und Statistik, Universität Konstanz, 78457 Konstanz, Germany, email: Juergen.Hausen@uni-konstanz.de
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Abstract

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We characterize embeddability of algebraic varieties into smooth toric varieties and prevarieties. Our embedding results hold also in an equivariant context and thus generalize a well-known embedding theorem of Sumihiro on quasiprojective $G$-varieties. The main idea is to reduce the embedding problem to the affine case. This is done by constructing equivariant affine conoids, a tool which extends the concept of an equivariant affine cone over a projective $G$-variety to a more general framework.

Keywords

14E2514C2014L3014M25
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 54 , Issue 3 , 01 June 2002 , pp. 554 - 570
Copyright
Copyright © Canadian Mathematical Society 2002

References

[1] A’Campo-Neuen, A. and Hausen, J., Quasi-projective reduction of toric varieties. Math. Z. (4) 233 (2000), 697–708.Google Scholar
[2] A’Campo-Neuen, A. and Hausen, J., Toric prevarieties and subtorus actions. Geom. Dedicata, mathAG/9912229, to appear.Google Scholar
[3] A’Campo-Neuen, A. and Hausen, J. and Schröer, S., Quotient presentations and homogeneous coordinates for toric varieties. Math. Nachr., math.AG/0005083, to appear.Google Scholar
[4] Białynicki-Birula, A., Finiteness of the number of maximal open subsets with good quotients. Transformation Groups (4) 3 (1998), 301–319.Google Scholar
[5] Borelli, M., Divisorial varieties. Pacific J. Math. 13 (1963), 375–388.Google Scholar
[6] Cox, D., The homogeneous coordinate ring of a toric variety. J. Algebraic Geom. 4 (1995), 17–51.Google Scholar
[7] Fulton, W., Introduction to Toric Varieties. Princeton University Press, Princeton, 1993.Google Scholar
[8] Hausen, J., On Włodarczyk's embedding theorem. Internat. J. Math. (6) 11 (2000), 811–836.Google Scholar
[9] Hausen, J. and Schröer, S., On embeddings into toric prevarieties. Preprint, math.AG/0002046.Google Scholar
[10] Humphreys, J. E., Linear Algebraic Groups. Graduate Texts in Math., Springer, Berlin-New York, 1975.Google Scholar
[11] Kajiwara, T., The functor of a toric variety with enough invariant effective cartier divisors. Tôhoku Math. J. (1) 50 (1998), 139–157.Google Scholar
[12] Knop, F., Kraft, H., Luna, D., Vust, T., Local properties of algebraic group actions. In: Algebraische Transformationsgruppen und Invariantentheorie, DMV Seminar Band 13. Birkhäuser, Basel, 1989.Google Scholar
[13] Sumihiro, H., Equivariant completion. J. Math. Kyoto Univ. 14 (1974), 1–28.Google Scholar
[14] Świeçicka, J., A Combinatorial construction of sets with good quotients by an action of a reductive group. Colloq. Math. (1) 87 (2001), 85–102.Google Scholar
[15] Włodarczyk, J., Embeddings in toric varieties and prevarieties. J. Algebraic Geom. 2 (1993), 705–726.Google Scholar