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

TL;DR

This paper classifies degree 2 semi-stable rational maps on the projective plane with large finite automorphism groups, establishing upper bounds on automorphism group sizes and providing a detailed conjugacy classification.

Contribution

It provides a complete classification of semi-stable degree 2 rational maps with finite automorphism groups of size at least 3, including explicit bounds and conjugacy class descriptions.

Findings

Automorphism group size is at most 24 for general maps.

Automorphism group size is at most 21 for morphisms.

Most maps have automorphism groups of size at most 6.

Abstract

Let $K$ be an algebraically closed field of characteristic $0$. In this paper we classify the $\text{PGL}_3(K)$-conjugacy classes of semi-stable dominant degree $2$ rational maps $f:{\mathbb P}^2_K\dashrightarrow{\mathbb P}^2_K$ whose automorphism group $$\text{Aut}(f):=\{\phi\in\text{PGL}_3(K): \phi^{-1}\circ f\circ\phi=f\}$$ is finite and of order at least $3$. In particular, we prove that $\#\text{Aut}(f)\le24$ in general, that $\#\text{Aut}(f)\le21$ for morphisms, and that $\#\text{Aut}(f)\le6$ for all but finitely many conjugacy classes of $f$.