an:07235509
Zbl 1448.37132
Doyle, John R.; Silverman, Joseph H.
Moduli spaces for dynamical systems with portraits
EN
Ill. J. Math. 64, No. 3, 375-465 (2020).
00452114
2020
j
37P45 37P15
endomorphisms; quotient moduli space; weights
Summary: A \textit{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.