Affine Lines on Affine Surfaces and the Makar–Limanov Invariant

Abstract

Abstract A smooth affine surface X defined over the complex field C is an ML 0 surface if the Makar– Limanov invariant ML( X ) is trivial. In this paper we study the topology and geometry of ML 0 surfaces. Of particular interest is the question: Is every curve C in X which is isomorphic to the affine line a fiber component of an A 1 -fibration on X ? We shall show that the answer is affirmative if the Picard number ρ( X ) = 0, but negative in case ρ( X ) ≥ 1. We shall also study the ascent and descent of the ML 0 property under proper maps.