On orbits of the automorphism group on an affine toric variety

Ivan Arzhantsev ; Ivan Bazhov

Open Mathematics, Tome 11 (2013), p. 1713-1724 / Harvested from The Polish Digital Mathematics Library

Résumé

Let X be an affine toric variety. The total coordinates on X provide a canonical presentation X¯→X of X as a quotient of a vector space X¯ by a linear action of a quasitorus. We prove that the orbits of the connected component of the automorphism group Aut(X) on X coincide with the Luna strata defined by the canonical quotient presentation.

Détail

Publié le : 2013-01-01

Zbl 1316.14097

EUDML-ID : urn:eudml:doc:269501

Bibliographie

[1] Arzhantsev I.V., Torsors over Luna strata, In: Torsors, Étale Homotopy and Applications to Rational Points, Edinburgh, January 10–14, 2011, London Math. Soc. Lecture Note Ser., 405, Cambridge University Press, Cambridge, 2013

[2] Arzhantsev I.V., Derenthal U., Hausen J., Laface A., Cox rings, preprint available at http://arxiv.org/abs/1003.4229

[3] Arzhantsev I.V., Flenner H., Kaliman S., Kutzschebauch F., Zaidenberg M., Flexible varieties and automorphism groups, Duke Math. J., 2013, 162(4), 767–823 http://dx.doi.org/10.1215/00127094-2080132[Crossref][WoS] | Zbl 1295.14057

[4] Arzhantsev I.V., Kuyumzhiyan K., Zaidenberg M., Flag varieties, toric varieties, and suspensions: three examples of infinite transitivity, Sb. Math., 2012, 203(7–8), 923–949 http://dx.doi.org/10.1070/SM2012v203n07ABEH004248[WoS][Crossref] | Zbl 1311.14059

[5] Arzhantsev I., Zaidenberg M., Acyclic curves and group actions on affine toric surfaces, In: Affine Algebraic Geometry, Osaka, March 3–6, 2011, World Scientific, Singapore, 2013, 1–41 http://dx.doi.org/10.1142/9789814436700_0001[Crossref] | Zbl 1319.14037

[6] Bazhov I., On orbits of the automorphism group on a complete toric variety, Beitr. Algebra Geom. (in press), DOI: 10.1007/s13366-011-0084-0 [Crossref] | Zbl 1327.14224

[7] Cox D.A., The homogeneous coordinate ring of a toric variety, J. Algebraic Geom., 1995, 4(1), 17–50 | Zbl 0846.14032

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

[9] Demazure M., Sous-groupes algébriques de rang maximum du groupe de Cremona, Ann. Sci. École Norm. Sup., 1970, 3(4), 507–588 | Zbl 0223.14009

[10] Freudenburg G., Algebraic Theory of Locally Nilpotent Derivations, Encyclopaedia Math. Sci., 136, Invariant Theory and Algebraic Transformation Groups, VII, Springer, Berlin, 2006 | Zbl 1121.13002

[11] Fulton W., Introduction to Toric Varieties, Ann. of Math. Stud., 131, Princeton University Press, Princeton, 1993 | Zbl 0813.14039

[12] Hausen J., Three lectures on Cox rings, In: Torsors, Étale Homotopy and Applications to Rational Points, Edinburgh, January 10–14, 2011, London Math. Soc. Lecture Note Ser., 405, Cambridge University Press, Cambridge, 2013, 3–60 http://dx.doi.org/10.1017/CBO9781139525350.002[Crossref]

[13] Humphreys J.E., Linear Algebraic Groups, Grad. Texts in Math., 21, Springer, New York-Heidelberg, 1975 http://dx.doi.org/10.1007/978-1-4684-9443-3[Crossref]

[14] Kuttler J., Reichstein Z., Is the Luna stratification intrinsic?, Ann. Inst. Fourier (Grenoble), 2008, 58(2), 689–721 http://dx.doi.org/10.5802/aif.2365[Crossref] | Zbl 1145.14047

[15] Luna D., Slices étales, In: Sur les Groupes Algébriques, Bull. Soc. Math. France Mém., 1973, 33, 81–105 | Zbl 0286.14014

[16] Oda T., Convex Bodies and Algebraic Geometry, Ergeb. Math. Grenzgeb., 15, Springer, Berlin, 1988

[17] Popov V.L., Vinberg E.B., Invariant theory, In: Algebraic Geometry, IV, Encyclopaedia Math. Sci., 55, Springer, Berlin, 1994, 123–284 http://dx.doi.org/10.1007/978-3-662-03073-8_2[Crossref]

[18] Ramanujam C.P., A note on automorphism group of algebraic variety, Math. Ann., 1964, 156(1), 25–33 http://dx.doi.org/10.1007/BF01359978[Crossref]