Effective cones of quotients of moduli spaces of stable 𝑛-pointed curves of genus zero

Rulla, William

doi:10.1090/S0002-9947-06-03851-7

Effective cones of quotients of moduli spaces of stable $n$-pointed curves of genus zero
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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