Moduli spaces of complex algebraic curves with isomorphic to \((\mathbb{Z}/2\mathbb{Z})^ m\) groups of automorphisms
Let \(K\) be a complex algebraic curve and \(G\) be a group of automorphisms of \(K\) which is isomorphic to \((\mathbb{Z}/2 \mathbb{Z})^m\). A full description of moduli spaces of pairs \((K,G)\) is given in this paper. Any connected component is described by the genus \(g\) of \(K/G\), nonnegative integers \(k_0\), \(k_1\) (where \(k_0 + 2k_1 \leq m\), \(m - k_0 \leq 2g)\) and a class of equivalence of functions \(S : (\mathbb{Z}/2 \mathbb{Z})^{k_0} \to \mathbb{Z}_+ = \{k \in \mathbb{Z} \mid k \geq 0\}\) relative to \(\Aut (\mathbb{Z}/2 \mathbb{Z})^{k_0}\). The component is a complex analytic space of the form \(\mathbb{C}^d/ \text{Mod}\), where Mod is a discrete group of holomorphic transformations. The problem is reduced to a description of classes of topological equivalence of a pair \((G,P)\), where \(G : P \to P\) is a group, isomorphic to \((\mathbb{Z}/2 \mathbb{Z})^m\), of autohomeomorphisms of an orientable surface \(P\).
scientific article; zbMATH DE number 743983