Explicit moduli of superelliptic curves with level structure (Q6619407)

The first main result of this article describes the connected components of the moduli \(\mathrm{Sup}_g^3[3]\) of trigonal superelliptic curves with level 3 structure, providing an explicit index set of the components, and identifies each component with the moduli of rational curves with a certain number of marked points. This result is a generalization of the known result by \textit{I. Dolgachev} and \textit{D. Ortland} [Point sets in projective spaces and theta functions. Paris: Société Mathématique de France (SMF) (1988; Zbl 0685.14029)] describing the irreducible components of the moduli of hyperelliptic curves with level 2 structure. The second result describes the connected components of the general moduli \(\mathrm{Sup}_g^p\) of (p-)superelliptic curves of genus g.