Moduli spaces for dynamical systems with portraits

Abstract

A portrait $\mathcal{P}$ on ${\mathbb{P}}^N$ is a pair of finite point sets $Y\subseteq X\subset {\mathbb{P}}^N$, a map $Y\to X$, and an assignment of weights to the points in $Y$. We construct a parameter space ${End}_d^N\left[\mathcal{P}\right]$ whose points correspond to degree $d$ endomorphisms $f:{\mathbb{P}}^N\to {\mathbb{P}}^N$ such that $f:Y\to X$ is as specified by a portrait $\mathcal{P}$, and prove the existence of the GIT quotient moduli space ${\mathcal{M}}_d^N\left[\mathcal{P}\right]:={End}_d^N//{SL}_{N+1}$ under the ${SL}_{N+1}$-action $(f,Y,X{)}^{\varphi }=({\varphi }^{-1}\circ f\circ \varphi ,{\varphi }^{-1}\left(Y\right),{\varphi }^{-1}\left(X\right))$ relative to an appropriately chosen line bundle. We also investigate the geometry of ${\mathcal{M}}_d^N\left[\mathcal{P}\right]$ and give two arithmetic applications.