On the variety of automorphisms of the affine plane

[摘要] The main subject of our study is GA(2,n), the variety of automophisms of the affine plane of degree bounded by a positive integer n. 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 GA(2,n) when n less than or equal to 9 and we show that GA(2,n) is reducible when n greater than or equal to 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 case n = 4. (C) 1997 Academic Press.

[发布日期] 1997-09-15