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Log canonical models for the moduli space of stable pointed rational curves


Author: Han-Bom Moon
Journal: Proc. Amer. Math. Soc. 141 (2013), 3771-3785
MSC (2010): Primary 14D20, 14E30, 14H10
DOI: https://doi.org/10.1090/S0002-9939-2013-11674-6
Published electronically: July 17, 2013
MathSciNet review: 3091767
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Abstract: We run Mori's program for the moduli space of stable pointed rational curves with divisor $ K +\sum a_{i}\psi _{i}$. We prove that the birational model for the pair is either the Hassett space of weighted pointed stable rational curves for the same weights or the GIT quotient of the product of projective lines with the linearization given by the same weights.


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  • [AGS10] V. Alexeev, A. Gibney and D. Swinarski. Conformal blocks divisors on $ \overline {M}_{0,n}$ from $ sl_2$. arXiv:1011.6659.
  • [AS08] V. Alexeev and D. Swinarski. Nef divisors on $ \overline {M}_{0,n}$ from GIT. arXiv:0812.0778.
  • [AC96] E. Arbarello and M. Cornalba. Combinatorial and algebro-geometric cohomology classes on the moduli spaces of curves. J. Algebraic Geom. 5 (1996), no. 4, 705-749. MR 1486986 (99c:14033)
  • [AC99] E. Arbarello and M. Cornalba. Calculating cohomology groups of moduli spaces of curves via algebraic geometry. Inst. Hautes Études Sci. Publ. Math. No. 88 (1998), 97-127 (1999). MR 1733327 (2001h:14030)
  • [Deb01] O. Debarre. Higher-dimensional algebraic geometry. Universitext, Springer-Verlag, New York, 2001. MR 1841091 (2002g:14001)
  • [FG03] G. Farkas and A. Gibney. The Mori cones of moduli spaces of pointed curves of small genus. Trans. Amer. Math. Soc. 355 (2003), no. 3, 1183-1199. MR 1938752 (2003m:14043)
  • [Fed10] M. Fedorchuk. Moduli spaces of weighted stable curves and log canonical models of $ \bar {M}_{g,n}$. Math. Res. Lett. 18 (2011), no. 4. MR 2831833
  • [FS08] M. Fedorchuk and D. Smyth. Ample divisors on moduli spaces of pointed rational curves. J. Algebraic Geom. 20 (2011), no. 4, 599-629. MR 2819671
  • [GG11] N. Giansiracusa and A. Gibney. The cone of type A, level one conformal blocks divisors. arXiv:1105.3139.
  • [GS10] N. Giansiracusa and M. Simpson. GIT compactifications of $ \overline {M}_{0,n}$ from conics. Inter. Math. Res. Notices 2011, no. 14, 3315-3334. MR 2817681 (2012j:14042)
  • [GKM02] A. Gibney, S. Keel and I. Morrison. Toward the ample cone of $ \overline {M}_{g,n}$. J. Amer. Math. Soc. 15 (2002), no. 2, 273-294. MR 1887636 (2003c:14029)
  • [HM98] J. Harris and I. Morrison. Moduli of curves. Graduate Texts in Mathematics, 187. Springer-Verlag, New York, 1998. MR 1631825 (99g:14031)
  • [Has03] B. Hassett. Moduli spaces of weighted pointed stable curves. Adv. Math. 173 (2003), no. 2, 316-352. MR 1957831 (2004b:14040)
  • [HK00] Y. Hu and S. Keel. Mori dream spaces and GIT. Michigan Math. J. 48 (2000), 331-348. MR 1786494 (2001i:14059)
  • [Kap93] M. Kapranov. Chow quotients of Grassmannians. I. I. M. Gelfand Seminar, 29-110, Adv. Soviet Math., 16, Part 2, Amer. Math. Soc., Providence, RI, 1993. MR 1237834 (95g:14053)
  • [Kee92] S. 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 (92f:14003)
  • [KMc96] S. Keel and J. McKernan. Contractible extremal rays on $ \overline {M}_{0,n}$. arXiv:9607009.
  • [KM11] Y.-H. Kiem and H.-B. Moon. Moduli spaces of weighted stable pointed rational curves via GIT. Osaka J. Math. 48 (2011), no. 4, 1115-1140. MR 2871297
  • [Kol90] J. Kollár. Projectivity of complete moduli. J. Differential Geom. 32 (1990), no. 1, 235-268. MR 1064874 (92e:14008)
  • [Knu83] F. Knudsen. The projectivity of the moduli space of stable curves. II. The stacks $ M_{g,n}$. Math. Scand. 52 (1983), no. 2, 161-199. MR 702953 (85d:14038a)
  • [Mo11] H.-B. Moon. Birational geometry of moduli spaces of curves of genus zero. Ph.D. dissertation, Seoul National University, 2011.
  • [MS11] I. Morrison and D. Swinarski. The $ S_{n}$-module structure of $ \mathrm {Pic}(\overline {M}_{0,n})$, preprint.
  • [Mum77] D. Mumford. Stability of projective varieties. Enseignement Math. (2) 23 (1977), no. 1-2, 39-110. MR 0450272 (56:8568)
  • [Pan97] R. Pandharipande. The canonical class of $ \overline {M}_{0,n}(\mathbb{P}^r, d)$. Internat. Math. Res. Notices 1997, no. 4, 173-186. MR 1436774 (98h:14067)
  • [Rul06] W. Rulla. Effective cones of quotients of moduli spaces of stable $ n$-pointed curves of genus zero. Trans. Amer. Math. Soc. 358 (2006), no. 7, 3219-3237. MR 2216265 (2007b:14053)
  • [Sim07] M. Simpson. On log canonical models of the moduli space of stable pointed curves. arXiv:0709.4037.

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Additional Information

Han-Bom Moon
Affiliation: Department of Mathematics, University of Georgia, Athens, Georgia 30602
Email: hbmoon@math.uga.edu

DOI: https://doi.org/10.1090/S0002-9939-2013-11674-6
Received by editor(s): October 3, 2011
Received by editor(s) in revised form: January 21, 2012
Published electronically: July 17, 2013
Communicated by: Lev Borisov
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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