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

Author: Atsushi Moriwaki
Journal: J. Amer. Math. Soc. 11 (1998), 569-600
MSC (1991): Primary 14H10, 14C20; Secondary 14G40
MathSciNet review: 1488349
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Abstract: 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 $1 \leq i \leq [g/2]$.

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Additional Information

Atsushi Moriwaki
Affiliation: Department of Mathematics, Faculty of Science, Kyoto University, Kyoto, 606-01, Japan

Keywords: Bogomolov's inequality, moduli space, stable curve
Received by editor(s): April 17, 1997
Received by editor(s) in revised form: January 2, 1998
Article copyright: © Copyright 1998 American Mathematical Society

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