MR 2831984 Reviewed Masuda T. Families of finite coverings of the Riemann sphere. Osaka J. Math. 48 (2011), no. 2, 515--540. (Reviewer Francesca Vetro) 14H30 (14H37)

Let $G$ be a finite group and let $H$ be a subgroup of $G$ which does not contain normal subgroups of $G$ except $\{ id \}$. The group $G$ acts on the set of the left coset of $G / H$ as follows: \begin{center} $(g, H a) \rightarrow H a g^{- 1}$. \end{center} The author observes that the action defined above is effective and this gives a permutation representation of $G$, $R: G \rightarrow S_{d}$, where $d =[G : H]$. The condition on $H$ ensures that $R$ is injective. Thus, $G$ can be seen as a transitive subgroup of $S_{d}$. Let $X$ and $ Y$ be connected complex varieties. A finite covering $f: X \rightarrow Y$, which branches at most at $B$, is said a $(G, H)-$coverings if there is a surjective homomorphism $\xi: \pi_{1} (Y - B, q_{0}) \rightarrow G$ satisfying the following conditions: \begin{itemize} \item[(1)] $R \, \xi$ is equivalent to the monodromy homomorphism of $f$, \item[(2)] $\xi^{-1} (H)$ corresponds to $f$.\end{itemize} In this paper, the author studies non-degenerate families of $(G, H)-$coverings of Riemann sphere, for a fixed $(G, H)$. In particular, he gives criterions for the existence or non-existence of Hurwitz families of special type of $(G, H)-$coverings of the Riemann sphere. The author calls a non-degenere family of $(G, H)-$coverings of Riemann sphere $I\hspace{-0.6ex}P^{1}$, a Hurwitz family if: \begin{itemize} \item[(i)] it contains all $(G, H)-$coverings, up to isomorphisms, which are topologically equivalent to a given $f_{0}: X_{0} \rightarrow I\hspace{-0.6ex}P^{1}$, \item[(ii)] any different members of it are not isomorphic. \end{itemize}