Moduli Spaces for Dynamical Systems with Portraits

Abstract

A $\text{portrait}$ $\mathcal{P}$ 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 ${\mathrm{End}}_d^N[\mathcal{P}]$ 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 ${\mathcal{M}}_d^N[\mathcal{P}]:={\mathrm{End}}_d^N//{\mathrm{SL}}_{N+1}$ under the ${\mathrm{SL}}_{N+1}$-action $(f,Y,X{)}^{\varphi }=({\varphi }^{-1}\circ f\circ \varphi ,{\varphi }^{-1}(Y),{\varphi }^{-1}(X))$ relative to an appropriately chosen line bundle. We also investigate the geometry of ${\mathcal{M}}_d^N[\mathcal{P}]$ and give two arithmetic applications.

91 pages