MR 2834249 Reviewed Hoshi Y., Galois-theoretic characterization of isomorphism classes of monodromically full hyperbolic curves of genus zero, Nagoya Math. J. (2011) 203, 47--100 ( Reviewer Francesca Vetro) 14H30 (14H10)

Let K be a finitely generated field of characteristic zero, l be a prime number and S be a scheme. In this paper, the author studies isomorphism classes of hyperbolic curves. The author calls the pair (C, D), where C is a scheme over S and D \subset C is a closed subscheme of C, a hyperbolic curve of type (g, r) over S if C is smooth and proper over S, if any geometric fiber of C \rightarrow S is a connected curve of genus g and if the composite D \rightarrow C \rightarrow S is a finite \'{e}tale covering over S of degree r. The main result of this paper is that the isomorphism class of a hyperbolic curve of genus zero over K that is l-monodromically full is completely determinated by the kernel of the natural pro-l outer Galois representation associated to the hyperbolic curve. The author moreover considers hyperbolic curves of type (0, 4) and gives sufficient conditions for such curves to be monodromically full.