The Mori cones of moduli spaces of pointed curves of small genus

Authors:
Gavril Farkas and Angela Gibney

Journal:
Trans. Amer. Math. Soc. **355** (2003), 1183-1199

MSC (2000):
Primary 14H10

DOI:
https://doi.org/10.1090/S0002-9947-02-03165-3

Published electronically:
November 7, 2002

MathSciNet review:
1938752

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Abstract | References | Similar Articles | Additional Information

Abstract: We compute the Mori cones of the moduli spaces of pointed stable curves of genus , when and are relatively small. For instance we show that for every curve in is equivalent to an effective combination of the components of the locus of curves with nodes. We completely describe the cone of nef divisors for the space , thus verifying Fulton's conjecture for this space. Using this description we obtain a classification of all the fibrations of .

**[AC]**E. Arbarello, M. Cornalba,*Calculating cohomology groups of moduli spaces of curves via algebraic geometry*, Inst. Hautes Études Sci. Publ. Math. No. 88(1998), 97-127. MR**2001h:14030****[Fa1]**C. Faber,*Intersection-theoretical computations on*, Parameter Spaces (Warsaw 1994), 71-81, Banach Center Publ. 36, 1996. MR**98j:14033****[Fa2]**C. Faber,*The nef cone of : a proof by inequalities only,*preprint.**[G]**A. Gibney,*Fibrations of*, Ph.D. Thesis, University of Texas, 2000.**[GKM]**A. Gibney, S. Keel, I. Morrison,*Towards the ample cone of*, J. Amer. Math. Soc. 15(2002), 273-294.**[HT]**B. Hassett, Y. Tschinkel,*On the effective cone of the moduli space of pointed rational curves*, math.AG/0110231.**[H]**B. Hunt,*The geometry of some special arithmetic quotients*, Lecture Notes in Mathematics 1637, Springer 1996. MR**98c:14033****[HMo]**J. Harris, I. Morrison,*Moduli of curves*, Springer, 1998. MR**99g:14031****[Kap]**M. Kapranov,*Veronese curves and the Grothendieck-Knudsen moduli space*, J. of Algebraic Geometry, 2 (1993), 239-262. MR**94a:14024****[Ke]**S. Keel,*Intersection theory on moduli spaces of -pointed curves of genus zero*, Trans. Amer. Math. Soc. 330(1992), 545-574. MR**92f:14003****[KMcK]**S. Keel, J. McKernan,*Contractible extremal rays on ,*math.AG/9607009.**[K]**J. Kollár,*Rational curves on algebraic varieties*, Springer 1996. MR**98c:14001****[Ve]**P. Vermeire,*A counterexample to Fulton's conjecture on*, J. of Algebra 248(2002), 780-784.

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Additional Information

**Gavril Farkas**

Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109-1109

Email:
gfarkas@umich.edu

**Angela Gibney**

Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109-1109

Email:
agibney@umich.edu

DOI:
https://doi.org/10.1090/S0002-9947-02-03165-3

Received by editor(s):
February 25, 2002

Published electronically:
November 7, 2002

Article copyright:
© Copyright 2002
American Mathematical Society