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2013 | 11 | 10 | 1713-1724
Tytuł artykułu

On orbits of the automorphism group on an affine toric variety

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Abstrakty
EN
Let X be an affine toric variety. The total coordinates on X provide a canonical presentation $$\bar X \to X$$ of X as a quotient of a vector space $$\bar X$$ 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.
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autor
  • Department of Higher Algebra, Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Leninskie Gory 1, GSP-1, Moscow, 119991, Russia, ibazhov@gmail.com
Bibliografia
  • [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]
  • [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]
  • [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]
  • [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]
  • [7] Cox D.A., The homogeneous coordinate ring of a toric variety, J. Algebraic Geom., 1995, 4(1), 17–50
  • [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
  • [10] Freudenburg G., Algebraic Theory of Locally Nilpotent Derivations, Encyclopaedia Math. Sci., 136, Invariant Theory and Algebraic Transformation Groups, VII, Springer, Berlin, 2006
  • [11] Fulton W., Introduction to Toric Varieties, Ann. of Math. Stud., 131, Princeton University Press, Princeton, 1993
  • [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]
  • [15] Luna D., Slices étales, In: Sur les Groupes Algébriques, Bull. Soc. Math. France Mém., 1973, 33, 81–105
  • [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]
Typ dokumentu
Bibliografia
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Identyfikator YADDA
bwmeta1.element.doi-10_2478_s11533-013-0273-1
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