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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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