Automorphism Groups and Invariant Theory on PN

2015/09/22 by João Alberto de Faria, de Faria, Joao Alberto, Benjamin Hutz +1
Computer Science · Mathematics · #13A50 #37P05 #37P45 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.1509.06670

openalex publication_date 2015/09/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let K be a field and f:ℙ^N → ℙ^N a morphism. There is a natural conjugation action on the space of such morphisms by elements of the projective linear group 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 PGL_N+1 determine endomorphisms of 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 ℙ¹ to endomorphisms of ℙ². A key component is an explicit bound on the size of the automorphism group depending on the degree of the endomorphism.

Citations

• Good Reduction of Periodic Points
• Cohen-Macaulay Rings, Invariant Theory, and the Generic Perfection of Determinantal Loci
• The Moduli Space of Cubic Rational Maps

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