will give in a moment. We do this in some detail, as it involves some subtleties. More on this can be found in the paper by Collas and Maugeais [@cm] which also addresses the question over which number field such a stack is defined.

Let $S$ be a fixed closed oriented surface of genus $g\ge 2$, $G$ a finite group and $\phi$ a monomorphism of $G$ in the mapping class group ${\operatorname{Mod}}(S)$ of $S$. By a theorem of Nielsen-Kerckhoff, we can lift $\phi$ to a monomorphism $\Phi$ of $G$ to a group of orientation preserving diffeomorphisms of $S$. This implies that $S$ admits a complex structure also invariant under $G$ so that we obtain a smooth projective complex curve $C$. We consider the $G$-curves which so arise. To be precise, we consider pairs $(C,G{\hookrightarrow}{\operatorname{Aut}}(C))$, where $C$ is a smooth projective complex algebraic curve $C$ of genus $\geq 2$ and $G{\hookrightarrow}{\operatorname{Aut}}(C)$ a monomorphism of groups such that there exists an orientation preserving $G$-equivariant homeomorphism of $C$ onto $S$. Observe that the last property only depends on the ${\operatorname{Mod}}(S)$-conjugacy class of $\phi$. So if we precompose $\phi$ with an automorphism $\sigma$ of $G$, then we get an isomorphic object if and only 