Towards the ample cone of
Authors:
Angela Gibney, Sean Keel and Ian Morrison
Journal:
J. Amer. Math. Soc. 15 (2002), 273294
MSC (2000):
Primary 14H10, 14E99
Published electronically:
December 20, 2001
MathSciNet review:
1887636
Fulltext PDF Free Access
Abstract 
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Abstract: In this paper we study the ample cone of the moduli space of stable pointed curves of genus . Our motivating conjecture is that a divisor on is ample iff it has positive intersection with all dimensional strata (the components of the locus of curves with at least nodes). This translates into a simple conjectural description of the cone by linear inequalities, and, as all the 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 . More precisely, there is a natural finite map whose image is the locus of curves with all components rational. Any strata either lies in or is numerically equivalent to a family of elliptic tails, and we show that a divisor is nef iff and is nef. We also give results on contractions (i.e. morphisms with connected fibers to projective varieties) of for 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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 , Calculating cohomology groups of moduli spaces of curves via algebraic geometry, Inst. Hautes Études Sci. Publ. Math. 88 (1998), 97127. MR 2001h:14030
 [CHM97]
 L. Caporaso, J. Harris, and B. Mazur, Uniformity of rational points, J. Amer. Math. Soc. 10 (1997), 135. MR 97d:14033
 [CornalbaHarris88]
 M. Cornalba and J. Harris, Divisor classes associated to families of stable varieties, with applications to the moduli space of curves, Ann. Sci. École Norm. Sup. (4) 21 (1988), 455475. MR 89j:14019
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 P. Deligne and D. Mumford, The irreduciblility of the space of curves of given genus, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 75109. MR 41:6850
 [Faber96]
 C. Faber, Intersectiontheoretical computations on , Banach Center Publ. 36, (1996), 7181. MR 98j:14033
 [Faber97]
 , Algorithms for computing intersection numbers on moduli spaces of curves, with application to the class of the locus of Jacobians, London Math. Soc. Lecture Note Ser., vol. 264, 1999, pp. 93109. MR 2000m:14032
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 A. Gibney, Fibrations of , Ph. D. Thesis, Univ. of Texas at Austin, 2000.
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 J. Harris and I. Morrison, Moduli of Curves, Grad. Texts in Math., vol. 187, SpringerVerlag, New York, 1998. MR 99g:14031
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 Y. Hu and S. Keel, Mori dream spaces and GIT, Michigan Math. J. 48 (2000), 331348. MR 2001i:14059
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 S. Keel, Intersection theory of moduli space of stable pointed curves of genus zero, Trans. Amer. Math. Soc. 330 (1992), 545574. MR 92f:14003
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 , Basepoint freeness for nef and big linebundles in positive characteristic, Annals of Math. (1999), 253286. MR 2000j:14011
 [KeelMcKernan96]
 S. Keel and J. McKernan, Contractible extremal rays of , preprint alggeom/9707016 (1996).
 [KollarMori98]
 J. Kollár and S. Mori, Birational geometry of algebraic varieties. With the collaboration of C. H. Clemens and A. Corti, Cambridge Tracts in Math., vol. 134, Cambridge University Press, Cambridge, 1998. MR 2000b:14018
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 M. Kontsevich and Y. Manin, Quantum cohomology of a product. With an appendix by R. Kaufmann, Invent. Math. 124 (1996), 313339. MR 97e:14064
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 A. Logan, Relations among divisors on the moduli space of curves with marked points, preprint math.AG/0003104, 2000.
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 A. Losev and Y. Manin, New Moduli Spaces of Pointed Curves and Pencils of Flat Connections, Michigan Math. J. 48 (2000), 443472. CMP 2001:03
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 A. Moriwaki, Relative Bogomolov's inequality and the cone of positive divisors on the moduli space of stable curves, J. Amer. Math. Soc. 11 (1998), 569600. MR 99a:14034
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 , Nef divisors in codimension one on the moduli space of stable curves, preprint math.AG/0005012, 2000.
 [Moriwaki01]
 , The Picard group of the moduli space of curves in positive characteristic, Internat. J. Math. 12 (2001), no. 5, 519534.
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 W. Rulla, Birational Geometry of , Ph D. Thesis, Univ. of Texas at Austin, 2000.
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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/S0894034701003848
PII:
S 08940347(01)003848
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
