  Let X be a smooth projective variety defined over an algebraically closed
field, and let Y in X be a reduced and irreducible ample divisor in X. We give
a numerical sufficient condition for a base point free pencil on $Y$ to be the
restriction of a base point free pencil on $X$. This result is then extended to
families of pencils and to morphisms to arbitrary smooth curves. Serrano had
already studied this problem in the case n=2 and 3, and Reider had then
attacked it in the case $n=2$ using vector bundle methods based on Bogomolov's
instability theorem on a surface (char(k)=0). The argument given here is based
on Bogomolov's theorem on an n-dimensional variety, and on its recent
adaptations to the setting of prime charachterstic (due to Shepherd-Barron and
Moriwaki).
