Rational connectedness and Galois covers of the projective line

Abstract

Let k be a p-adic field. Some time ago, D. Harbater [9] proved that any finite group G may be realized as a regular Galois group over the rational function field in one variable k(t), namely there exists a finite field extension $F/k(t)$, Galois with group G, such that F is a regular extension of k (i.e. k is algebraically closed in F). Moreover, one may arrange that a given k-place of k(t) be totally split in F. Harbater proved this theorem for k an arbitrary complete valued field. Rather formal arguments ([10, \S 4.5]; \S2 hereafter) then imply that the theorem holds over any `large' field k. This in turn is a special case of a result of Pop [15], hence will be referred to as the Harbater/Pop theorem. We refer to [10], [16], [6] for precise references to the literature (work of D\`ebes, Deschamps, Fried, Haran, Harbater, Jarden, Liu, Pop, Serre, and V\"olklein). Most proofs (see [10], [19, 8.4.4, p.~93] and Liu's contribution to [16]; see however [15]) first use direct arguments to establish the theorem when G is a cyclic group (here the nature of the ground field is irrelevant), then proceed by patching, using either formal or rigid geometry, together with GAGA theorems. In the present paper, where I take the case of algebraically closed fields for granted, I show how a technique recently developed by Koll\'ar [12] may be used to give a quite different proof of the Harbater/Pop theorem, when the `large' field k has characteristic zero. This proof actually gives more than the original result (see comment after statement of Theorem 1).