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Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)

 

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


Authors: Angela Gibney, Sean Keel and Ian Morrison
Journal: J. Amer. Math. Soc. 15 (2002), 273-294
MSC (2000): Primary 14H10, 14E99
Published electronically: December 20, 2001
MathSciNet review: 1887636
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Abstract: 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.


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Additional Information

Angela Gibney
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
Email: agibney@math.lsa.umich.edu

Sean Keel
Affiliation: Department of Mathematics, University of Texas at Austin, Austin, Texas 78712
Email: keel@fireant.ma.utexas.edu

Ian Morrison
Affiliation: Department of Mathematics, Fordham University, Bronx, New York 10458
Email: morrison@fordham.edu

DOI: http://dx.doi.org/10.1090/S0894-0347-01-00384-8
PII: S 0894-0347(01)00384-8
Keywords: Ample cone, Mori cone, moduli space, stable curve
Received by editor(s): September 5, 2000
Published electronically: December 20, 2001
Additional Notes: During this research, the first two authors received partial support from a Big XII faculty research grant, and a grant from the Texas Higher Education Coordinating Board.
The first author also received partial support from the Clay Mathematics Institute, and the second from the NSF
The third author’s research was partially supported by a Fordham University Faculty Fellowship and by grants from the Centre de Recerca Matemática for a stay in Barcelona and from the Consiglio Nazionale di Ricerche for stays in Pisa and Genova
Dedicated: To Bill Fulton on his sixtieth birthday
Article copyright: © Copyright 2001 American Mathematical Society