Towards the ample cone of \overline{𝑀}_{𝑔,𝑛}

Gibney, Angela¹ ; Keel, Sean² ; Morrison, Ian³

¹ Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
² Department of Mathematics, University of Texas at Austin, Austin, Texas 78712
³ Department of Mathematics, Fordham University, Bronx, New York 10458

Journal of the American Mathematical Society, Tome 15 (2002) no. 2, pp. 273-294

Résumé

In this paper we study the ample cone of the moduli space $\overline {M}_{g,n}$ of stable $n$-pointed curves of genus $g$. Our motivating conjecture is that a divisor on $\overline {M}_{g,n}$ is ample iff it has positive intersection with all $1$-dimensional strata (the components of the locus of curves with at least $3g+n-2$ nodes). This translates into a simple conjectural description of the cone by linear inequalities, and, as all the $1$-strata are rational, includes the conjecture that the Mori cone is polyhedral and generated by rational curves. Our main result is that the conjecture holds iff it holds for $g=0$. More precisely, there is a natural finite map $r: \overline {M}_{ 0, 2g+n} \rightarrow \overline {M}_{g,n}$ whose image is the locus $\overline {R}_{g,n}$ of curves with all components rational. Any $1$-strata either lies in $\overline {R}_{g,n}$ or is numerically equivalent to a family $E$ of elliptic tails, and we show that a divisor $D$ is nef iff $D \cdot E \geq 0$ and $r^{*}(D)$ is nef. We also give results on contractions (i.e. morphisms with connected fibers to projective varieties) of $\overline {M}_{g,n}$ for $g \geq 1$ showing that any fibration factors through a tautological one (given by forgetting points) and that the exceptional locus of any birational contraction is contained in the boundary.

Détail

DOI : 10.1090/S0894-0347-01-00384-8

Bibliographie

[1] Arbarello, Enrico, Cornalba, Maurizio Combinatorial and algebro-geometric cohomology classes on the moduli spaces of curves J. Algebraic Geom. 1996 705 749

[2] Arbarello, Enrico, Cornalba, Maurizio Calculating cohomology groups of moduli spaces of curves via algebraic geometry Inst. Hautes Études Sci. Publ. Math. 1998

[3] Caporaso, Lucia, Harris, Joe, Mazur, Barry Uniformity of rational points J. Amer. Math. Soc. 1997 1 35

[4] 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

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

[6] Faber, Carel Intersection-theoretical computations on \overline{ℳ}_{ℊ} 1996 71 81

[7] Faber, Carel Algorithms for computing intersection numbers on moduli spaces of curves, with an application to the class of the locus of Jacobians 1999 93 109

[8] Harris, Joe, Morrison, Ian Moduli of curves 1998

[9] Hu, Yi, Keel, Sean Mori dream spaces and GIT Michigan Math. J. 2000 331 348

[10] Keel, Sean Intersection theory of moduli space of stable 𝑛-pointed curves of genus zero Trans. Amer. Math. Soc. 1992 545 574

[11] Keel, Seã¡N Basepoint freeness for nef and big line bundles in positive characteristic Ann. of Math. (2) 1999 253 286

[12] Kollã¡R, Jã¡Nos, Mori, Shigefumi Birational geometry of algebraic varieties 1998

[13] Kontsevich, M., Manin, Yu. Quantum cohomology of a product Invent. Math. 1996 313 339

[14] Moriwaki, Atsushi Relative Bogomolov’s inequality and the cone of positive divisors on the moduli space of stable curves J. Amer. Math. Soc. 1998 569 600