# GIT Stability of Henon Maps

## TL;DR

This paper analyzes the geometric invariant theory (GIT) stability of generalized Henon maps in projective space, revealing conditions under which these maps are unstable, semistable, or not stable, and classifying unstable cases.

## Contribution

It provides a comprehensive GIT stability classification of Henon maps in various dimensions and degrees, including a full classification of unstable maps in the case of degree 2 and dimension 2.

## Key findings

- Henon maps are GIT unstable if N≥3 or d≥3.
- They are semistable but not stable when N=d=2.
- Complete classification of unstable maps in Rat_2^2.

## Abstract

In this paper we study the locus of generalized degree $d$ Henon maps in the parameter space $\operatorname{Rat}_d^N$ of degree $d$ rational maps $\mathbb{P}^N\to\mathbb{P}^N$ modulo the conjugation action of $\operatorname{SL}_{N+1}$. We show that Henon maps are in the GIT unstable locus if $N\ge3$ or $d\ge3$, and that they are semistable, but not stable, in the remaining case of $N=d=2$. We also give a general classification of all unstable maps in $\operatorname{Rat}_2^2$.