  Let X be a smooth projective curve of genus g \geq 2 over an algebraically
closed field k of characteristic p > 0. Let M_X be the moduli space of
semistable rank-2 vector bundles over X with trivial determinant. The relative
Frobenius map F: X \to X_1 induces by pull-back a rational map V: M_{X_1} \to
M_{X}. In this paper we show the following results. 1) For any line bundle L
over X, the rank-p vector bundle F_*L is stable. 2) The rational map V has base
points, i.e., there exist stable bundles E over X_1 such that F^* E is not
semistable. 3) Let B \subset M_{X_1} denote the scheme-theoretical base locus
of V. If g=2, p>2 and X ordinary, then B is a 0-dimensional local complete
intersection of length {2/3}p(p^2 -1) and the degree of V equals {1/3}p(p^2
+2).
