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

Let $f : X \to Y$ be a surjective and projective morphism of smooth quasi-projective 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 $y \in Y$, then $f_*\left( 2rc_2(E) - (r-1)c_1(E)^2 \right)$ is weakly positive at $y$. We apply this result to obtain the following description of the cone of weakly positive ${\mathbb{Q}}$-Cartier divisors on the moduli space of stable curves. Let $\overline{\mathcal{M}}_g$ (resp. $\mathcal{M}_g$) be the moduli space of stable (resp. smooth) curves of genus $g \geq 2$. Let $\lambda$ be the Hodge class, and let the $\delta _i$'s ($i = 0, \ldots, [g/2]$) be the boundary classes. Then, a ${\mathbb{Q}}$-Cartier divisor $x \lambda + \sum _{i=0}^{[g/2]} y_i \delta _i$ on $\overline{\mathcal{M}}_g$ is weakly positive over $\mathcal{M}_g$ if and only if $x \geq 0$, $g x + (8g + 4) y_0 \geq 0$, and $i(g-i) x + (2g+1) y_i \geq 0$ for all $1 \leq i \leq [g/2]$.