Reduction of covers and Hurwitz spaces
نویسندگان
چکیده
In this paper we study the reduction of Galois covers of curves, from characteristic zero to positive characteristic. The starting point is a recent result of Raynaud, which gives a criterion for good reduction for covers of the projective line branched at three points. We use the ideas of Raynaud to study the case of covers of the projective line branched at four points. Under some condition on the Galois group, we generalize the criterion for good reduction of Raynaud. As a new ingredient, we use the Hurwitz space of such covers. Combining our results on reduction of covers with the Hurwitz space approach, we are able to describe the reduction of the Hurwitz space modulo p and compute the number of covers with good reduction. 2000 Mathematical Subject Classification: 14H30, 14G32
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