Plane polynomial automorphisms of fixed multidegree
type: Item
instance of: scholarly article
title: Plane polynomial automorphisms of fixed multidegree
zbMATH Open document ID: 1187.14064
author: Jean-Philippe Furter
published in: Mathematische Annalen
publication date: 2009-03-26T00:00:00Z
review text: 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 \(G_{\leq m}\).
reviewed by: Jérémy Blanc
Mathematics Subject Classification ID: 14R10
Mathematics Subject Classification ID: 14J50
Mathematics Subject Classification ID: 14E07
zbMATH DE Number: 5533130
zbMATH Keywords: polynomial automorphisms
zbMATH Keywords: algebraic groups of infinite dimension
zbMATH Keywords: multidegree
MaRDI profile type: Publication
full work available at URL: s00208-008-0296-2
OpenAlex ID: W2002272035
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DOI: 10.1007/S00208-008-0296-2
description: scientific article; zbMATH DE number 5533130