  Given a geometrically irreducible subscheme X in P^n over F_q of dimension at
least 2, we prove that the fraction of degree d hypersurfaces H such that the
intersection of H and X is geometrically irreducible tends to 1 as d tends to
infinity. We also prove variants in which X is over an extension of F_q, and in
which the immersion of X in P^n is replaced by a more general morphism.
