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Automorphisms of affine surfaces with A1-fibrations

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 Hi(X;Q) = (0) for i > 0 thenX is called a logQ-homology plane (ifX is smooth then X is simply called a Q-homology plane). Let Ga 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 A1-fibration. Of particular interest are surfaces that admit a regular action of Ga. Such actions up to conjugacy correspond in a bijective manner to A1-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 (X) that are killed under all the locally nilpotent derivations of (X) is called the Makar-Limanov invariant of X and denoted by ML(X). If a smooth affine surface has two independent Ga actions then its MakarLimanov invariant is trivial. Gizatullin [9] and Bertin [2] gave a necessary and sufficient condition for this to happen. More recently, Bandman and MakarLimanov [1] proved that a smooth affine surface X with trivial canonical bundle and ML(X) = C is an affine surface in A3 defined by {xy = p(z)}, where p(z) is a polynomial with distinct roots. Masuda and Miyanishi [12] applied this to determine the structure of a Q-homology plane with trivial ML-invariant. They proved that such a surface is a quotient of the Bandman–Makar-Limanov hypersurface by the action of a finite cyclic group (see result (3) in the listing that follows). In this paper we extend the last result to the case of log Q-homology planes in Section 2. Similar and related results in Section 2 and Section 3 have been obtained independently by Daigle and Russell [4] and Dubouloz [5]. An automorphism of a smooth affine surface sends fibers of one A1-fibration with affine base to the fibers of another A1-fibration. If these two fibrations are different then the Makar-Limanov invariant of the surface is trivial. If a smooth affine surface has an A1-fibration whose base is not an affine curve, then this fibration does not correspond to a Ga action. In this case the geometry of the fibration enters into the picture. In Section 4 we give a sufficient condition for uniqueness of an A1fibration on a smooth affine surface. This involves the number of multiple fibers