Flexibility of normal affine horospherical varieties
HTML articles powered by AMS MathViewer
- by Sergey Gaifullin and Anton Shafarevich PDF
- Proc. Amer. Math. Soc. 147 (2019), 3317-3330 Request permission
Abstract:
We investigate flexibility of affine varieties with an action of a linear algebraic group. Flexibility of a smooth affine variety with only constant invertible functions and a locally transitive action of a reductive group is proved. Also we show that a normal affine complexity-zero horospherical variety with only constant invertible functions is flexible.References
- Ivan Arzhantsev, Ulrich Derenthal, Jürgen Hausen, and Antonio Laface, Cox rings, Cambridge Studies in Advanced Mathematics, vol. 144, Cambridge University Press, Cambridge, 2015. MR 3307753
- I. V. Arzhantsev and S. A. Gaĭfullin, Cox rings, semigroups, and automorphisms of affine varieties, Mat. Sb. 201 (2010), no. 1, 3–24 (Russian, with Russian summary); English transl., Sb. Math. 201 (2010), no. 1-2, 1–21. MR 2641086, DOI 10.1070/SM2010v201n01ABEH004063
- Ivan Arzhantsev and Sergey Gaifullin, The automorphism group of a rigid affine variety, Math. Nachr. 290 (2017), no. 5-6, 662–671. MR 3636369, DOI 10.1002/mana.201600295
- I. Arzhantsev, H. Flenner, S. Kaliman, F. Kutzschebauch, and M. Zaidenberg, Flexible varieties and automorphism groups, Duke Math. J. 162 (2013), no. 4, 767–823. MR 3039680, DOI 10.1215/00127094-2080132
- I. V. Arzhantsev, M. G. Zaĭdenberg, and K. G. Kuyumzhiyan, Flag varieties, toric varieties, and suspensions: three examples of infinite transitivity, Mat. Sb. 203 (2012), no. 7, 3–30 (Russian, with Russian summary); English transl., Sb. Math. 203 (2012), no. 7-8, 923–949. MR 2986429, DOI 10.1070/SM2012v203n07ABEH004248
- Victor Batyrev and Fatima Haddad, On the geometry of $\textrm {SL}(2)$-equivariant flips, Mosc. Math. J. 8 (2008), no. 4, 621–646, 846 (English, with English and Russian summaries). MR 2499357, DOI 10.17323/1609-4514-2008-8-4-621-646
- F. Donzelli, Makar-Limanov invariants, Derksen invariants, flexible points, arXiv:1107.3340, 2011.
- Hubert Flenner and Mikhail Zaidenberg, On the uniqueness of $\Bbb C^*$-actions on affine surfaces, Affine algebraic geometry, Contemp. Math., vol. 369, Amer. Math. Soc., Providence, RI, 2005, pp. 97–111. MR 2126657, DOI 10.1090/conm/369/06807
- S. Gaifullin, Automorphisms of Danielewski varieties, arXiv:1709.09237, 2017.
- Friedrich Knop, Über Hilberts vierzehntes Problem für Varietäten mit Kompliziertheit eins, Math. Z. 213 (1993), no. 1, 33–36 (German). MR 1217668, DOI 10.1007/BF03025706
- L. Makar-Limanov, On the group of automorphisms of a surface $x^ny=P(z)$, Israel J. Math. 121 (2001), 113–123. MR 1818396, DOI 10.1007/BF02802499
- A. L. Onishchik and È. B. Vinberg, Lie groups and algebraic groups, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1990. Translated from the Russian and with a preface by D. A. Leites. MR 1064110, DOI 10.1007/978-3-642-74334-4
- V. L. Popov, Quasihomogeneous affine algebraic varieties of the group $\textrm {SL}(2)$, Izv. Akad. Nauk SSSR Ser. Mat 37 (1973), 792–832 (Russian). MR 0340263
- È. B. Vinberg and V. L. Popov, A certain class of quasihomogeneous affine varieties, Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 749–764 (Russian). MR 0313260
- I. R. Shafarevich (ed.), Algebraic geometry. IV, Encyclopaedia of Mathematical Sciences, vol. 55, Springer-Verlag, Berlin, 1994. Linear algebraic groups. Invariant theory; A translation of Algebraic geometry. 4 (Russian), Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1989 [ MR1100483 (91k:14001)]; Translation edited by A. N. Parshin and I. R. Shafarevich. MR 1309681, DOI 10.1007/978-3-662-03073-8
- A. A. Shafarevich, Flexibility of $S$-varieties of semisimple groups, Mat. Sb. 208 (2017), no. 2, 121–148 (Russian, with Russian summary); English transl., Sb. Math. 208 (2017), no. 1-2, 285–310. MR 3608041, DOI 10.4213/sm8625
- Dmitry A. Timashev, Homogeneous spaces and equivariant embeddings, Encyclopaedia of Mathematical Sciences, vol. 138, Springer, Heidelberg, 2011. Invariant Theory and Algebraic Transformation Groups, 8. MR 2797018, DOI 10.1007/978-3-642-18399-7
Additional Information
- Sergey Gaifullin
- Affiliation: Faculty of Mechanics and Mathematics, Department of Higher Algebra, Lomonosov Moscow State University, Leninskie Gory 1, Moscow, 119991 Russia; Faculty of Computer Science, National Research University Higher School of Economics, Kochnovskiy Proezd 3, Moscow, 125319 Russia
- MR Author ID: 838217
- Email: sgayf@yandex.ru
- Anton Shafarevich
- Affiliation: Faculty of Mechanics and Mathematics, Department of Higher Algebra, Lomonosov Moscow State University, Leninskie Gory 1, Moscow, 119991 Russia; Faculty of Computer Science, National Research University Higher School of Economics, Kochnovskiy Proezd 3, Moscow, 125319 Russia
- MR Author ID: 1205633
- Email: shafarevich.a@gmail.com
- Received by editor(s): May 16, 2018
- Received by editor(s) in revised form: November 30, 2018
- Published electronically: April 8, 2019
- Additional Notes: The first author was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics “BASIS”
- Communicated by: Jerzy Weyman
- © Copyright 2019 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 147 (2019), 3317-3330
- MSC (2010): Primary 13N15, 14J50; Secondary 14R20, 13A50
- DOI: https://doi.org/10.1090/proc/14528
- MathSciNet review: 3981110