Automorphism Groups and Invariant Theory on PN

Abstract

Let $K$ be a field and $f:{\mathbb{P}}^N\to {\mathbb{P}}^N$ a morphism. There is a natural conjugation action on the space of such morphisms by elements of the projective linear group ${\text{PGL}}_{N+1}$. The group of automorphisms, or stabilizer group, of a given $f$ for this action is known to be a finite group. In this article, we address two mainly computational problems concerning automorphism groups. Given a finite subgroup of ${\text{PGL}}_{N+1}$ determine endomorphisms of ${\mathbb{P}}^N$ with that group as subgroup of its automorphism group. In particular, we show that every finite subgroup occurs infinitely often and discuss some associated rationality problems. Inversely, given an endomorphism determine its automorphism group. In particular, we extended the Faber-Manes-Viray fixed-point algorithm for ${\mathbb{P}}^1$ to endomorphisms of ${\mathbb{P}}^2$. A key component is an explicit bound on the size of the automorphism group depending on the degree of the endomorphism.

correction to bound on size of automorphism group in terms of degree of the map

References in corpus (3)

• The Moduli Space of Cubic Rational Maps
• Uniform bounds for pre-periodic points in families of twists
• Almost Newton, sometimes Lattès

Cited by in corpus (1)

• A classification of degree $2$ semi-stable rational maps ${\mathbb{P}}^2\to {\mathbb{P}}^2$ with large finite dynamical automorphism group