Automorphisms of Affine Surfaces with A~1-Fibrations

摘要

Let X be a normal affine surface defined over the complex field C, which has at worst quotient singularities. We call X simply a log affine surface. If further H_i(X; Q) = (0) for i > 0 then X is called a log Q-homology plane (if X is smooth then X is simply called a Q-homology plane). Let G_a denote the complex numbers with addition as an algebraic group. In this paper we are mainly interested in log affine surfaces X that have an A~1-fibration. Of particular interest are surfaces that admit a regular action of G_a. Such actions up to conjugacy correspond in a bijective manner to A~1-fibrations on X with base a smooth affine curve. Algebraically, these actions correspond bijectively to locally nilpotent derivations of the coordinate ring ?(X) of X. The set of all elements of Γ(X) that are killed under all the locally nilpotent derivations of Γ(X) is called the Makar-Limanov invariant of X and denoted by ML(X).