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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Relative Bogomolov’s inequality and the cone of positive divisors on the moduli space of stable curves
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by Atsushi Moriwaki PDF
J. Amer. Math. Soc. 11 (1998), 569-600 Request permission

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
  • 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