Automorphism groups of rigid affine surfaces: The identity component

Abstract

It is known that the identity component of the automorphism group of a projective algebraic variety is an algebraic group.This is not true in general for quasi-projective varieties.In this note, we address the question as to when, given an affine algebraic surface Y , the identity component Aut • (Y ) of the automorphism group Aut(Y ) is an algebraic group.We show that this occurs if and only if Y admits no effective action of the additive group G a of the field.In the latter case, Aut • (Y ) is an algebraic torus of rank at most 2.