The Michigan Mathematical Journal

Polyhedral divisors and SL2-actions on affine T-varieties

Ivan Arzhantsev and Alvaro Liendo

Full-text: Open access

Article information

Michigan Math. J., Volume 61, Issue 4 (2012), 731-762.

First available in Project Euclid: 16 November 2012

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 13N15: Derivations 14L30: Group actions on varieties or schemes (quotients) [See also 13A50, 14L24, 14M17]
Secondary: 14M25: Toric varieties, Newton polyhedra [See also 52B20] 14R20: Group actions on affine varieties [See also 13A50, 14L30] 14M17: Homogeneous spaces and generalizations [See also 32M10, 53C30, 57T15]


Arzhantsev, Ivan; Liendo, Alvaro. Polyhedral divisors and SL 2 -actions on affine T -varieties. Michigan Math. J. 61 (2012), no. 4, 731--762. doi:10.1307/mmj/1353098511.

Export citation


  • K. Altmann and J. Hausen, Polyhedral divisors and algebraic torus actions, Math. Ann. 334 (2006), 557–607.
  • K. Altmann, J. Hausen, and H. Süss, Gluing affine torus actions via divisorial fans, Transform. Groups 13 (2008), 215–242.
  • K. Altmann, N. Owen Ilten, L. Petersen, H. Süss, and R. Vollmert, The geometry of T-varieties, Contributions to algebraic geometry (P. Pragacz, ed.), IMPANGA Lecture Notes (to appear).
  • I. V. Arzhantsev, On $\text{\rm SL}_{2}$-actions of complexity one, Izv. Ross. Akad. Nauk Ser. Mat. 61 (1997), 3–18 (Russian); English translation in Izv. Math. 61 (1997), 685–698.
  • V. Batyrev and F. Haddad, On the geometry of $\text{\rm SL}(2)$-equivariant flips, Mosc. Math. J. 8 (2008), 621–646, 846.
  • F. Berchtold and J. Hausen, Demushkin's theorem in codimension one, Math. Z. 244 (2003), 697–703.
  • M. Demazure, Sous-groupes algébriques de rang maximum du groupe de Cremona, Ann. Sci. École Norm. Sup. (4) 3 (1970), 507–588.
  • –––, Anneaux gradués normaux, Introduction à la théorie des singularités, II, Travaux en Cours, 37, pp. 35–68, Hermann, Paris, 1988.
  • I. Dolgachev, Lectures on invariant theory, London Math. Soc. Lecture Note Ser., 296, Cambridge Univ. Press, Cambridge, 2003.
  • H. Flenner and M. Zaidenberg, Locally nilpotent derivations on affine surfaces with a ${\Bbb C}^{*}$-action, Osaka J. Math. 42 (2005), 931–974.
  • G. Freudenburg, Algebraic theory of locally nilpotent derivations, Encyclopaedia Math. Sci., 136, Springer-Verlag, Berlin, 2006.
  • W. Fulton, Introduction to toric varieties, Ann. of Math. Stud., 131, Princeton Univ. Press, Princeton, NJ, 1993.
  • S. A. Gaĭfullin, Affine toric $\text{\rm SL}(2)$-embeddings, Mat. Sb. (N.S.) 199 (2008), 3–24.
  • G. Kempf, F. Knudsen, D. Mumford, and B. Saint-Donat, Toroidal embeddings. I, Lecture Notes in Math., 339, Springer-Verlag, Berlin, 1973.
  • H. Kraft, Geometrische Methoden in der Invariantentheorie, Aspects Math., D1, Vieweg, Braunschweig, 1984.
  • A. Liendo, Affine ${\Bbb T}$-varieties of complexity one and locally nilpotent derivations, Transform. Groups 15 (2010), 389–425.
  • –––, ${\Bbb G}_{\text{\rm a}}$-actions of fiber type on affine ${\Bbb T}$-varieties, J. Algebra 324 (2010), 3653–3665.
  • D. Luna, Sur les orbites fermées des groupes algébriques réductifs, Invent. Math. 16 (1972), 1–5.
  • –––, Adhérences d'orbite et invariants, Invent. Math. 29 (1975), 231–238.
  • V. L. Popov, Quasihomogeneous affine algebraic varieties of the group $\text{\rm SL}(2),$ Izv. Akad. Nauk SSSR Ser. Mat. 37 (1973), 792–832.
  • –––, Contractions of actions of reductive algebraic groups, Mat. Sb. (N.S.) 130 (1986), 310–334.
  • D. A. Timashev, Classification of $G$-manifolds of complexity 1, Izv. Ross. Akad. Nauk Ser. Mat. 61 (1997), 127–162 (Russian); English translation in Izv. Math. 61 (1997), 363–397.
  • –––, Torus actions of complexity one, Toric topology, Contemp. Math., 460, pp. 349–364, Amer. Math. Soc., Providence, RI, 2008.