26th Autumn School in Algebraic Geometry

C+ and finite groups actions on affine spaces. Locally nilpotent derivations.

Lukecin, Poland, September 14h - September 21th, 2003

Teachers: Daniel Daigle (Ottawa), Gene Freudenburg (Evansville)

Theory of G_a-actions on affine varieties, locally nilpotent derivations, with special attention to the case of the affine space A^n for small values of n (n=3,4 are particularly intriguing). Connections of this subject with other questions, such as Hilbert's Fourteenth Problem, polynomial automorphisms or the classification of surfaces.


  • Locally nilpotent derivations, by Daniel Daigle
  • Notes to lectures of Gene Freudenburg

    Further readings:

  • A. van den Essen, "Polynomial Automorphisms and the Jacobian Conjecture", Birkhaauser Verlag, Progress in Mathematics, vol. 190 (Basel, Boston, Berlin) 2000
  • V.L. Popov, "On actions of G_a on A^n", in: Springer Verlag, Lect. Notes in Math., vol. 1271 (Berlin, Heidelberg, New York) 1987
  • D. Snow, "Unipotent actions on affine space", in: Birkh\"auser Verlag, Progress in Math., vol.80 (Basel, Boston, Berlin) 1989
  • H. Bass, "A non-triangluar action of G_a on A^3", J. Pure Appl. Algebra 33 (1984) 1-5
  • J. Deveney, D. Finston, "Algebraic aspects of additive group actions on complex affine space", in: Kluwer Academic Publishers, "Automorphisms of Affine Spaces" (The Netherlands) 1995
  • J. Humphreys, "Hilbert's Fourteenth Problem", Amer. Math. Monthly 70 (1978) 341-353
  • O. Hadas, L. Makar-Limanov, "Newton polytopes of constants of locally nilpotent derivations", Comm. Algebra 28 (2000) 3667-3678
  • C. S. Seshadri, "On a theorem of Weitzenbock in invariant theory", J. Math. Kyoto Univ. 1 (1962) 403-409
  • A. Tyc, "An elementary proof of the Weitzenbock theorem", Colloq. Math. 78 (1998) 123-132
  • L. Tan, "An algorithm for explicit generatoirs of the invariants of the basci G_a actions", Comm. Algebra 17 (1989) 565-572
  • P. Roberts, "An infinitely generated symbolic blow-up in a power series ring and a new counterexample to Hilbert's fourteenth problem", J. Algebra 132 (1990) 461-473
  • D. Daigle, "Homogeneous locally nilpotent derivations of k[x,y,z]", J. Pure Appl. Algebra 128
  • D. Daigle, G. Freudenburg, "A counterexample to Hilbert's fourteenth problem in dimension five", J. Algebra J. Algebra 221 (1999) 528-535
  • D. Daigle, G. Freudenburg, "Locally nilpotent derivations over a UFD and an application to rank two locally nilpotent derivations on k[x1,...,xn]", J. Algebra 204 (1998) 353-371
  • G. Freudenburg, "Local slice constructions in k[x,y,z]", J. Pure Appl. Algebra 128 (1998) 109-132
    Some of the above articles are available at Daniel Daigle's web page

    Organizers: Mariusz Koras and Jaroslaw Buczynski.

    A joint picture of all participants

    The school was financially supported by Institute of Mathematics of Warsaw University, by Polish State Committee for Scientific Research and by EAGER (European Algebraic Geometry Research Training Network).

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