Plane polynomial automorphisms of fixed multidegree

Abstract

Let \({\mathcal{G}}\) be the group of polynomial automorphisms of the complex affine plane. On one hand, \({\mathcal {G}}\) can be endowed with the structure of an infinite dimensional algebraic group (see Shafarevich in Math USSR Izv 18:214–226, 1982) and on the other hand there is a partition of \({\mathcal{G}}\) according to the multidegree (see Friedland and Milnor in Ergod Th Dyn Syst 9:67–99, 1989). Let \({{\mathcal{G}}_d}\) denote the set of automorphisms whose multidegree is equal to d. We prove that \({{\mathcal{G}}_d}\) is a smooth, locally closed subset of \({\mathcal{G}}\) and show some related results. We give some applications to the study of the varieties \({{\mathcal G}_{= \, m}}\) (resp. \({{\mathcal G}_{\leq \, m}}\)) of automorphisms whose degree is equal to m (resp. is less than or equal to m).