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

Michelle Manes; Joseph H. Silverman

doi:10.5802/afst.1614

A classification of degree

2

semi-stable rational maps

ℙ^{2} \to ℙ^{2}

with large finite dynamical automorphism group

Michelle Manes ¹; Joseph H. Silverman ²

¹ Department of Mathematics, University of Hawaii, Honolulu, HI 96822, USA
² Mathematics Department, Box 1917 Brown University, Providence, RI 02912, USA

Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 28 (2019) no. 4, pp. 733-811.

Abstract
Abstract

Let $K$ be an algebraically closed field of characteristic $0$ . In this paper we classify the ${PGL}_{3} (K)$ -conjugacy classes of semi-stable dominant degree $2$ rational maps $f : ℙ_{K}^{2} ⤏ ℙ_{K}^{2}$ whose automorphism group

Aut (f) : = {φ \in {PGL}_{3} (K) : φ^{- 1} \circ f \circ φ = f}

is finite and of order at least $3$ . In particular, we prove that $# Aut (f) \leq 24$ in general, that $# Aut (f) \leq 21$ for morphisms, and that $# Aut (f) \leq 6$ for all but finitely many conjugacy classes of $f$ .

Soit $K$ un corps algébriquement clos de charactéristique $0$ . Dans cet article nous classifions les ${PGL}_{3} (K)$ -classes de conjugaison de fonctions rationelles $f : ℙ_{K}^{2} ⤏ ℙ_{K}^{2}$ de degré $2$ dominantes et semi-stables dont le groupe d’automorphismes

Aut (f) : = {φ \in {PGL}_{3} (K) : φ^{- 1} \circ f \circ φ = f}

est fini et d’ordre au moins $3$ . En particulier, nous démontrons que $# Aut (f) \leq 24$ en général, que $# Aut (f) \leq 21$ pour les morphismes et que $# Aut (f) \leq 6$ pour toutes excepté un nombre fini de classes de conjugaisons de $f$ .

Received: 2016-07-25
Accepted: 2017-07-24
Published online: 2019-12-09

DOI: 10.5802/afst.1614

Classification: 37P45, 37P05
Keywords: dynamical moduli space

Author's affiliations:

Michelle Manes ¹; Joseph H. Silverman ²

¹ Department of Mathematics, University of Hawaii, Honolulu, HI 96822, USA
² Mathematics Department, Box 1917 Brown University, Providence, RI 02912, USA

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     title = {A classification of degree~$2$ semi-stable rational maps $\protect \mathbb{P}^2\rightarrow \protect \mathbb{P}^2$ with large finite dynamical automorphism group},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {733--811},
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Michelle Manes; Joseph H. Silverman. A classification of degree $2$ semi-stable rational maps $\protect \mathbb{P}^2\rightarrow \protect \mathbb{P}^2$ with large finite dynamical automorphism group. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 28 (2019) no. 4, pp. 733-811. doi : 10.5802/afst.1614. https://afst.centre-mersenne.org/articles/10.5802/afst.1614/

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