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

Moriwaki, Atsushi¹

¹ Department of Mathematics, Faculty of Science, Kyoto University, Kyoto, 606-01, Japan

Journal of the American Mathematical Society, Tome 11 (1998) no. 3, pp. 569-600

Résumé

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 $1 \leq i \leq [g/2]$.

Détail

DOI : 10.1090/S0894-0347-98-00261-6

Bibliographie

[1] Bogomolov, F. A. Holomorphic tensors and vector bundles on projective manifolds Izv. Akad. Nauk SSSR Ser. Mat. 1978

[2] Bost, Jean-Benoã®T Semi-stability and heights of cycles Invent. Math. 1994 223 253

[3] Cornalba, Maurizio, Harris, Joe Divisor classes associated to families of stable varieties, with applications to the moduli space of curves Ann. Sci. École Norm. Sup. (4) 1988 455 475

[4] Deligne, P., Mumford, D. The irreducibility of the space of curves of given genus Inst. Hautes Études Sci. Publ. Math. 1969 75 109

[5] Fulton, William Intersection theory 1984

[6] Gieseker, David Stable vector bundles and the Frobenius morphism Ann. Sci. École Norm. Sup. (4) 1973 95 101

[7] Gieseker, D. On a theorem of Bogomolov on Chern classes of stable bundles Amer. J. Math. 1979 77 85

[8] Hartshorne, Robin Ample vector bundles on curves Nagoya Math. J. 1971 73 89

[9] Jouanolou, Jean-Pierre Théorèmes de Bertini et applications 1983

[10] Knudsen, Finn Faye, Mumford, David The projectivity of the moduli space of stable curves. I. Preliminaries on “det” and “Div” Math. Scand. 1976 19 55

[11] Maruyama, Masaki The theorem of Grauert-Mülich-Spindler Math. Ann. 1981 317 333

[12] Miyaoka, Yoichi The Chern classes and Kodaira dimension of a minimal variety 1987 449 476

[13] Moriwaki, Atsushi Arithmetic Bogomolov-Gieseker’s inequality Amer. J. Math. 1995 1325 1347

[14] Moriwaki, Atsushi A sharp slope inequality for general stable fibrations of curves J. Reine Angew. Math. 1996 177 195

[15] Mumford, David Stability of projective varieties Enseign. Math. (2) 1977 39 110

[16] Paranjape, Kapil, Ramanan, S. On the canonical ring of a curve 1988 503 516

[17] Xiao, Gang Fibered algebraic surfaces with low slope Math. Ann. 1987 449 466

[18] Zhang, Shouwu Admissible pairing on a curve Invent. Math. 1993 171 193