Generalized superelliptic Riemann surfaces

A closed Riemann surface $\mathcal{X}$, of genus $g\ge 2$, is called a generalized superelliptic curve of level $n\ge 2$ if it admits an order $n$ conformal automorphism $\tau$ so that $\mathcal{X}\mathrm{/}⟨\tau ⟩$ has genus zero and $\tau$ is central in $\mathrm{A}\mathrm{u}\mathrm{t}(\mathcal{X})$; the cyclic group $H=⟨\tau ⟩$ is called a generalized superelliptic group of level $n$ for $\mathcal{X}$. These Riemann surfaces are natural generalizations of hyperelliptic Riemann surfaces (when $n=2$). We provide an algebraic curve description of these Riemann surfaces in terms of their groups of automorphisms. Also, we observe that the generalized superelliptic group $H$ of level $n$ is unique, with the exception of a very particular family of exceptional generalized superelliptic Riemann surfaces for $n$ even. In particular, the uniqueness holds if either: (i) $n$ is odd or (ii) the quotient $\mathcal{X}\mathrm{/}H$ has all its cone points of order $n$ (for instance, when $\mathcal{X}$ is a superelliptic curve of level $n$). In the non-exceptional case, we use this uniqueness property of its generalized superelliptic group $H$ to observe that the corresponding curves are definable over their fields of moduli if $\mathrm{A}\mathrm{u}\mathrm{t}(\mathcal{X})\mathrm{/}H$ is neither trivial or cyclic.