# 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)

Risultato della ricerca: Review articlepeer review

## Abstract

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.
Lingua originale English 0 MATHEMATICAL REVIEWS 2013 Published - 2013

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