GIT Stability of Henon Maps

Abstract

In this paper we study the locus of generalized degree $d$ Henon maps in the parameter space ${\mathrm{Rat}}_d^N$ of degree $d$ rational maps ${\mathbb{P}}^N\to {\mathbb{P}}^N$ modulo the conjugation action of ${\mathrm{SL}}_{N+1}$. We show that Henon maps are in the GIT unstable locus if $N\ge 3$ or $d\ge 3$, 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 ${\mathrm{Rat}}_2^2$.

arXiv comment9 pages