Monodromy Groups of Coverings of Curves

Summary

We consider finite separable coverings of curves f : X → Y over a field of characteristic p ≥ 0. We are interested in describing the possible monodromy groups of this cover if the genus of X is fixed. There has been much progress on this problem over the past decade in characteristic zero. Recently Frohardt and Magaard completed the final step in resolving the Guralnick-Thompson conjecture showing that only finitely many nonabelian simple groups other than alternating groups occur as composition factors for a fixed genus. There is an ongoing project to get a complete list of the monodromy groups of indecomposable rational functions with only tame ramification. In this article, we focus on positive characteristic. There are more possible groups but we show that many simple groups do not occur as composition factors for a fixed genus. We also give a reduction theorem reducing the problem to the case of almost simple groups. We also obtain some results on bounding the size of automorphism groups of curves in positive characteristic and discuss the relationship with the first problem. We note that prior to these results there was not a single example of a finite simple group which could be ruled out as a composition factor of the monodromy group of a rational function in any positive characteristic.

1. Introduction

Let k be a perfect field of characteristic p ≥ 0. Suppose that X and Y are smooth projective curves over k and f : X → Y is a nonconstant separable rational map.