Moduli Spaces for Dynamical Systems with Portraits

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