On the Tame Fundamental Groups of Curves over Algebraically Closed Fields of Characteristic > 0

Summary

We prove that the isomorphism class of the tame fundamental group of a smooth, connected curve over an algebraically closed field k of characteristic p > 0 determines the genus g and the number n of punctures of the curve, unless (g, n) = (0,0), (0,1). Moreover, assuming g = 0, n > 1, and that k is the algebraic closure of the prime field 𝔽_p , we prove that the isomorphism class of the tame fundamental group even completely determines the isomorphism class of the curve as a scheme (though not necessarily as a fc-scheme). As a key tool to prove these results, we generalize Raynaud's theory of theta divisors.

Introduction

Let k be an algebraically closed field of characteristic p > 0, and U a smooth, connected curve over k. (A curve is a separated scheme of dimension 1.) We denote by X the smooth compactification of U and put S = X — U. We define non-negative integers g and n to be the genus of X and the cardinality of the point set S, respectively.

In [T2], we proved that the isomorphism class of the (profinite) fundamental group π₁ (U) of U determines the pair (g, n), and that, when g = 0 and k is the algebraic closure of the prime field 𝔽_p, the isomorphism class of π₁ (U) even completely determines the isomorphism class of the curve as a scheme.