Journal of the Mathematical Society of Japan

The tame and the wild automorphisms of an affine quadric threefold

Stéphane LAMY and Stéphane VÉNÉREAU

Full-text: Open access


We generalize the notion of a tame automorphism to the context of an affine quadric threefold and we prove that there exist non-tame automorphisms.

Article information

J. Math. Soc. Japan, Volume 65, Number 1 (2013), 299-320.

First available in Project Euclid: 24 January 2013

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14R20: Group actions on affine varieties [See also 13A50, 14L30]

tame automorphisms affine quadric


LAMY, Stéphane; VÉNÉREAU, Stéphane. The tame and the wild automorphisms of an affine quadric threefold. J. Math. Soc. Japan 65 (2013), no. 1, 299--320. doi:10.2969/jmsj/06510299.

Export citation


  • I. V. Arzhantsev and S. A. Gaĭfullin, Cox rings, semigroups, and automorphisms of affine varieties, Mat. Sb., 201 (2010), 3–24.
  • W. Fulton and J. Harris, Representation Theory, A first course, Readings in Mathematics, Grad. Texts in Math., 129, Springer-Verlag, New York, 1991.
  • S. Kaliman and L. Makar-Limanov, On the Russell-Koras contractible threefolds, J. Algebraic Geom., 6 (1997), 247–268.
  • S. Kuroda, A generalization of the Shestakov-Umirbaev inequality, J. Math. Soc. Japan, 60 (2008), 495–510.
  • S. Kuroda, Shestakov-Umirbaev reductions and Nagata's conjecture on a polynomial automorphism, Tohoku Math. J. (2), 62 (2010), 75–115.
  • S. Lamy, Sur la structure du groupe d'automorphismes de certaines surfaces affines, Publ. Mat., 49 (2005), 3–20.
  • S. Lang, Algebra, third edition, Grad. Texts in Math., 211, Springer-Verlag, New York, 2002.
  • L. Makar-Limanov, On groups of automorphisms of a class of surfaces, Israel J. Math., 69 (1990), 250–256.
  • L. Makar-Limanov and J.-T. Yu, Degree estimate for subalgebras generated by two elements, J. Eur. Math. Soc. (JEMS), 10 (2008), 533–541.
  • I. P. Shestakov and U. U. Umirbaev, Poisson brackets and two-generated subalgebras of rings of polynomials, J. Amer. Math. Soc., 17 (2004), 181–196 (electronic).
  • I. P. Shestakov and U. U. Umirbaev, The tame and the wild automorphisms of polynomial rings in three variables, J. Amer. Math. Soc., 17 (2004), 197–227 (electronic).
  • S. Vénéreau, A parachute for the degree of a polynomial in algebraically independent ones, Math. Ann., 349 (2011), 589–597.
  • M. Zaĭdenberg, Exotic algebraic structures on affine spaces, Algebra i Analiz, 11 (1999), 3–73 (translation in St. Petersburg Math. J., 11 (2000), 703–760).