Effective cones of quotients of moduli spaces of stable $n$-pointed curves of genus zero
HTML articles powered by AMS MathViewer
- by William F. Rulla PDF
- Trans. Amer. Math. Soc. 358 (2006), 3219-3237 Request permission
Abstract:
Let $X_n := \overline {M}_{0,n}$, the moduli space of $n$-pointed stable genus zero curves, and let $X_{n,m}$ be the quotient of $X_n$ by the action of $\mathcal {S}_{n-m}$ on the last $n-m$ marked points. The cones of effective divisors $\overline {NE}^1(X_{n,m})$, $m = 0,1,2$, are calculated. Using this, upper bounds for the cones $Mov(X_{n,m})$ generated by divisors with moving linear systems are calculated, $m = 0,1$, along with the induced bounds on the cones of ample divisors of $\overline {M}_g$ and $\overline {M}_{g,1}$. As an application, the cone $\overline {NE}^1(\overline {M}_{2,1})$ is analyzed in detail.References
- P. Deligne and D. Mumford, The irreducibility of the space of curves of given genus, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 75–109. MR 262240, DOI 10.1007/BF02684599
- 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. 93–109. MR 1714822, DOI 10.1017/CBO9780511721540.006
- Yi Hu and Sean Keel, Mori dream spaces and GIT, Michigan Math. J. 48 (2000), 331–348. Dedicated to William Fulton on the occasion of his 60th birthday. MR 1786494, DOI 10.1307/mmj/1030132722
- Joe Harris and Ian Morrison, Moduli of curves, Graduate Texts in Mathematics, vol. 187, Springer-Verlag, New York, 1998. MR 1631825
- 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. 83–96. MR 1941624, DOI 10.1090/conm/314/05424
- Sean Keel, Intersection theory of moduli space of stable $n$-pointed curves of genus zero, Trans. Amer. Math. Soc. 330 (1992), no. 2, 545–574. MR 1034665, DOI 10.1090/S0002-9947-1992-1034665-0
- Seán Keel, Basepoint freeness for nef and big line bundles in positive characteristic, Ann. of Math. (2) 149 (1999), no. 1, 253–286. MR 1680559, DOI 10.2307/121025
- Seán Keel and James McKernan, Contractible extremal rays on $\overline {M}_{0,n}$, Preprint, alg-geom/9607009, July 1996.
- 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, DOI 10.1017/CBO9780511662560
- 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, Springer-Verlag, Berlin, 1996. MR 1440180, DOI 10.1007/978-3-662-03276-3
- Atsushi Moriwaki, The $\Bbb Q$-Picard group of the moduli space of curves in positive characteristic, Internat. J. Math. 12 (2001), no. 5, 519–534. MR 1843864, DOI 10.1142/S0129167X01000964
- Rahul Pandharipande, The canonical class of $\overline {M}_{0,n}(\mathbf P^r,d)$ and enumerative geometry, Internat. Math. Res. Notices 4 (1997), 173–186. MR 1436774, DOI 10.1155/S1073792897000123
- PORTA, POlyhedron Representation Transformation Algorithm, avaliable at http://www.iwr.uni-heidelberg.de/groups/comopt/software/PORTA.
- William Rulla, The birational geometry of $\overline {M}_3$ and $\overline {M}_{2,1}$, Ph.D. thesis, University of Texas at Austin, 2001.
- Peter Vermeire, A counterexample to Fulton’s conjecture on $\overline M_{0,n}$, J. Algebra 248 (2002), no. 2, 780–784. MR 1882122, DOI 10.1006/jabr.2001.9044
Additional Information
- William F. Rulla
- Affiliation: Department of Mathematics, University of Georgia, Athens, Georgia 30602
- Email: rulla@math.uga.edu
- 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.
- © Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 358 (2006), 3219-3237
- MSC (2000): Primary 14E05, 14H10; Secondary 14E30
- DOI: https://doi.org/10.1090/S0002-9947-06-03851-7
- MathSciNet review: 2216265