Effective cones of quotients of moduli spaces of stable pointed curves of genus zero
Author:
William F. Rulla
Journal:
Trans. Amer. Math. Soc. 358 (2006), 32193237
MSC (2000):
Primary 14E05, 14H10; Secondary 14E30
Published electronically:
February 20, 2006
MathSciNet review:
2216265
Fulltext PDF Free Access
Abstract 
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Abstract: Let , the moduli space of pointed stable genus zero curves, and let be the quotient of by the action of on the last marked points. The cones of effective divisors , , are calculated. Using this, upper bounds for the cones generated by divisors with moving linear systems are calculated, , along with the induced bounds on the cones of ample divisors of and . As an application, the cone is analyzed in detail.
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 Carel Faber, Algorithms for computing intersection numbers on moduli spaces of curves, with an application to the class of the locus of Jacobians, New trends in algebraic geometry (Warwick, 1996), London Math. Soc. Lecture Note Ser., vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 93109. MR 1714822 (2000m:14032)
 [HK00]
 Yi Hu and Sean Keel, Mori dream spaces and GIT, Michigan Math. J. 48 (2000), 331348, Dedicated to William Fulton on the occasion of his 60th birthday. MR 1786494 (2001i:14059)
 [HM98]
 Joe Harris and Ian Morrison, Moduli of curves, Graduate Texts in Mathematics, vol. 187, SpringerVerlag, New York, 1998. MR 1631825 (99g:14031)
 [HT02]
 Brendan Hassett and Yuri Tschinkel, On the effective cone of the moduli space of pointed rational curves, Topology and geometry: commemorating SISTAG, Contemp. Math., vol. 314, Amer. Math. Soc., Providence, RI, 2002, pp. 8396. MR 1941624 (2004d:14028)
 [Kee92]
 Seán Keel, Intersection theory of moduli space of stable pointed curves of genus zero, Trans. Amer. Math. Soc. 330 (1992), no. 2, 545574. MR 1034665 (92f:14003)
 [Kee99]
 , Basepoint freeness for nef and big line bundles in positive characteristic, Ann. of Math. (2) 149 (1999), no. 1, 253286. MR 1680559 (2000j:14011)
 [KM96]
 Seán Keel and James McKernan, Contractible extremal rays on , Preprint, alggeom/9607009, July 1996.
 [KM98]
 János Kollár and Shigefumi Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, vol. 134, Cambridge University Press, Cambridge, 1998, With the collaboration of C. H. Clemens and A. Corti, Translated from the 1998 Japanese original. MR 1658959 (2000b:14018)
 [Kol96]
 János Kollár, Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 32, SpringerVerlag, Berlin, 1996. MR 1440180 (98c:14001)
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 Atsushi Moriwaki, The Picard group of the moduli space of curves in positive characteristic, Internat. J. Math. 12 (2001), no. 5, 519534. MR 1843864 (2002j:14033)
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 Peter Vermeire, A counterexample to Fulton's conjecture on , J. Algebra 248 (2002), no. 2, 780784. MR 1882122 (2002k:14043)
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Additional Information
William F. Rulla
Affiliation:
Department of Mathematics, University of Georgia, Athens, Georgia 30602
Email:
rulla@math.uga.edu
DOI:
http://dx.doi.org/10.1090/S0002994706038517
PII:
S 00029947(06)038517
Keywords:
Moduli space,
rational curve,
birational geometry,
classification of morphisms/rational maps
Received by editor(s):
December 5, 2003
Received by editor(s) in revised form:
September 9, 2004
Published electronically:
February 20, 2006
Additional Notes:
This paper is a product of a VIGRE seminar on $\overline{M}_{0,n}$ conducted by V. Alexeev at the University of Georgia, Athens, during the Spring of 2002. Thanks to S. Keel for posing the question motivating the paper, and to him, R. Varley, and E. Izadi for help and advice. Thanks also to the referee for many valuable comments. PORTA was used in calculating several examples. Xfig was used for the figures.
Article copyright:
© Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
