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

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- by Atsushi Moriwaki
- J. Amer. Math. Soc.
**11**(1998), 569-600 - DOI: https://doi.org/10.1090/S0894-0347-98-00261-6
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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 $1 \leq i \leq [g/2]$.## References

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## Bibliographic Information

**Atsushi Moriwaki**- Affiliation: Department of Mathematics, Faculty of Science, Kyoto University, Kyoto, 606-01, Japan
- Email: moriwaki@kusm.kyoto-u.ac.jp
- Received by editor(s): April 17, 1997
- Received by editor(s) in revised form: January 2, 1998
- © Copyright 1998 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**11**(1998), 569-600 - MSC (1991): Primary 14H10, 14C20; Secondary 14G40
- DOI: https://doi.org/10.1090/S0894-0347-98-00261-6
- MathSciNet review: 1488349