Plane polynomial automorphisms of fixed multidegree
Let \(G\) be the group of polynomial automorphisms of the complex affine plane. There are two well-known structures on \(G\). On one hand, the embedding of \(G\) into the set of endomorphisms of the plane, isomorphic to \(\mathbb{C}[X]^2\), gives a structure of an algebraic group of infinite dimension to \(G\). On the other hand, \(G\) is the amalgamated product of the group of affine automorphism and of the group of triangular automorphisms. With this second structure, any element \(g\) of \(G\) has a multidegree, which is a \(n\)-uple of integers \((d_1,\dots,d_n)\) with \(d_i\geq 2\) for each \(i\). These are the degrees of the triangular automorphisms which appear in a minimal decomposition of \(G\) in the amalgamated product. Observe that the degree of \(g\) is the product of the \(d_i\). If \(d\) is a \(n\)-tuple, we denote by \(G_d\) the set of all elements of \(G\) having multidegree \(d\). Moreover, for any positive degree, the subset \(G_{\leq d}\) of all elements of \(G\) of degree \(\leq d\) is know to be a closed subvariety of \(G\) of finite dimension. In this article, the authors proves the following results: Theorem A: If \(d=(d_1,\dots,d_l)\) and \(m\) is the product of the \(d_i\), then \(G_d\) is closed in \(G_{\leq m}\). Theorem B: Each \(G_d\) is a smooth, locally closed subset of \(G\). The author also gives corollaries and other results concerning the varieties \(G_d\) and \(G_{\leq m}\).
scientific article; zbMATH DE number 5533130
scientific article