Relative Bogomolov's inequality and the cone of positive divisors on the moduli space of stable curves

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].