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Relative Bogomolov’s inequality and the cone of positive divisors on the moduli space of stable curves. (English) Zbl 0893.14004
Summary: 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 $dimf=1$. 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 ${f}_{*}\left(2r{c}_{2}\left(E\right)-\left(r-1\right){c}_{1}{\left(E\right)}^{2}\right)$ is weakly positive at $y$. We apply this result to obtain the following description of the cone of weakly positive $ℚ$-Cartier divisors on the moduli space of stable curves. Let ${\overline{ℳ}}_{g}$ (resp. ${ℳ}_{g}$) be the moduli space of stable (resp. smooth) curves of genus $g\ge 2$. Let $\lambda$ be the Hodge class, and let the ${\delta }_{i}$’s ($i=0,...,\left[g/2\right]$) be the boundary classes. Then, a $ℚ$-Cartier divisor $x\lambda +{\sum }_{i=0}^{\left[g/2\right]}{y}_{i}{\delta }_{i}$ on ${\overline{ℳ}}_{g}$ is weakly positive over ${ℳ}_{g}$ if and only if $x\ge 0$, $gx+\left(8g+4\right){y}_{0}\ge 0$, and $i\left(g-i\right)x+\left(2g+1\right){y}_{i}\ge 0$ for all $1\le i\le \left[g/2\right]$.

MSC:
 14H10 Families, algebraic moduli (curves) 14C20 Divisors, linear systems, invertible sheaves 14G40 Arithmetic varieties and schemes; Arakelov theory; heights 57R20 Characteristic classes and numbers (differential topology)