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Moduli spaces for dynamical systems with portraits. (English) Zbl 1448.37132
Summary: 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 $$\operatorname{End}_d^N[\mathcal{P}]$$ 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 $$\mathcal{M}_d^N[\mathcal{P}]:=\operatorname{End}_d^N/\!/\operatorname{SL}_{N+1}$$ under the $$\operatorname{SL}_{N+1}$$-action $$(f,Y,X)^{\phi}=(\phi^{-1}\circ{f}\circ \phi ,\phi^{-1}(Y),\phi^{-1}(X))$$ relative to an appropriately chosen line bundle. We also investigate the geometry of $$\mathcal{M}_d^N[\mathcal{P}]$$ and give two arithmetic applications.

MSC:
 37P45 Families and moduli spaces in arithmetic and non-Archimedean dynamical systems 37P15 Dynamical systems over global ground fields
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References:
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