  Let f : X --> Y be a projective morphism of smooth algebraic varieties over
an algebraically closed field of characteristic zero with dim f = 1. Let E be a
vector bundle of rank r on X. In this paper, we would like to show that if X_y
is smooth and E_y is semistable for some point y of Y, then f_* (2r c_2(E) -
(r-1) c_1(E)^2) is weakly positive at y. We apply this result to obtain the
following description of the cone of weakly positive $\QQ$-Cartier divisors on
the moduli space of stable curves. Let M_g (resp. M_g^0) be the moduli space of
stable (resp. smooth) curves of genus g >= 2. Let h be the Hodge class and
d_i's (i = 0,...,[g/2]) the boundary classes. A Q-Cartier divisor x h + y_0 d_0
+ ... + y_[g/2] d_[g/2] is weakly positive over M_g^0 if and only if x >= 0, g
x + (8g + 4) y_0>= 0, and i(g-i) x + (2g+1) y_i>= 0 for all 1 <= i <= [g/2].
