On the specialisation homomorphism of fundamental groups for algebraic curves in characteristic p>0
by
Mohamed Saidi
University of Bonn, Germany
Coauthors: Florian Pop (Bonn)

Let M_g --> bar F_p be the moduli space of curves of genus g > 1. For any point x of M_g one can associate a geometric fundamental group \pi₁(x), which is defined up to isomorphism. If y is a point of M_g which specializes in a point x, Grothendieck defined a continuous surjective homomorphism between the corresponding fundamental groups \pi₁(y) --> > \pi₁(x). We are interested in the question whether the above homomorphism is an isomorphism. Our main result is that there exists an infinite set X of closed points of M_g, which has positive density, and such that for every point x in X the above specialisation map can not be an isomorphism. We also discuss some applications of this result.

Date received: February 8, 1999