On the Variety of Automorphisms of the Affine Plane

Abstract

The main subject of our study is GA2,n, the variety of automophisms of the affine plane of degree bounded by a positive integern. After detailing some definitions and notations in Section 1, we give in Section 2 an algorithm to decide whether an endomorphism of the affine plane over an integral domain is a tame automorphism. Then, by applying this algorithm to the Nagata automorphism, we recover easily its known results. In Section 6, we compute the number of irreducible components of GA2,nwhenn≤9 and we show that GA2,nis reducible whenn≥4. Our proofs are based on a precise decomposition theorem for automorphisms given in Section 3 and a characterization of length one automorphisms given in Section 5. Finally, in Section 7, we give some details on the casen=4.