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

Moon, Han-Bom

doi:10.1090/S0002-9939-2013-11674-6

Log canonical models for the moduli space of stable pointed rational curves
HTML articles powered by AMS MathViewer

by Han-Bom Moon PDF

Proc. Amer. Math. Soc. 141 (2013), 3771-3785 Request permission

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.

References

V. Alexeev, A. Gibney and D. Swinarski. Conformal blocks divisors on $\overline {M}_{0,n}$ from $sl_2$. arXiv:1011.6659.
V. Alexeev and D. Swinarski. Nef divisors on $\overline {M}_{0,n}$ from GIT. arXiv:0812.0778.
Enrico Arbarello and Maurizio Cornalba, Combinatorial and algebro-geometric cohomology classes on the moduli spaces of curves, J. Algebraic Geom. 5 (1996), no. 4, 705–749. MR 1486986
Enrico Arbarello and Maurizio Cornalba, Calculating cohomology groups of moduli spaces of curves via algebraic geometry, Inst. Hautes Études Sci. Publ. Math. 88 (1998), 97–127 (1999). MR 1733327
Olivier Debarre, Higher-dimensional algebraic geometry, Universitext, Springer-Verlag, New York, 2001. MR 1841091, DOI 10.1007/978-1-4757-5406-3
Gavril Farkas and Angela 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, DOI 10.1090/S0002-9947-02-03165-3
Maksym Fedorchuk, Moduli of weighted pointed stable curves and log canonical models of $\overline {\scr M}_{g,n}$, Math. Res. Lett. 18 (2011), no. 4, 663–675. MR 2831833, DOI 10.4310/MRL.2011.v18.n4.a6
Maksym Fedorchuk and David Ishii Smyth, Ample divisors on moduli spaces of pointed rational curves, J. Algebraic Geom. 20 (2011), no. 4, 599–629. MR 2819671, DOI 10.1090/S1056-3911-2011-00547-X
N. Giansiracusa and A. Gibney. The cone of type A, level one conformal blocks divisors. arXiv:1105.3139.
Noah Giansiracusa and Matthew Simpson, GIT compactifications of $\scr M_{0,n}$ from conics, Int. Math. Res. Not. IMRN 14 (2011), 3315–3334. MR 2817681, DOI 10.1093/imrn/rnq228
Angela Gibney, Sean Keel, and Ian Morrison, Towards the ample cone of $\overline M_{g,n}$, J. Amer. Math. Soc. 15 (2002), no. 2, 273–294. MR 1887636, DOI 10.1090/S0894-0347-01-00384-8
Joe Harris and Ian Morrison, Moduli of curves, Graduate Texts in Mathematics, vol. 187, Springer-Verlag, New York, 1998. MR 1631825
Brendan Hassett, Moduli spaces of weighted pointed stable curves, Adv. Math. 173 (2003), no. 2, 316–352. MR 1957831, DOI 10.1016/S0001-8708(02)00058-0
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
M. M. Kapranov, Chow quotients of Grassmannians. I, I. M. Gel′fand Seminar, Adv. Soviet Math., vol. 16, Amer. Math. Soc., Providence, RI, 1993, pp. 29–110. MR 1237834
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
S. Keel and J. McKernan. Contractible extremal rays on $\overline {M}_{0,n}$. arXiv:9607009.
Young-Hoon Kiem and Han-Bom Moon, Moduli spaces of weighted pointed stable rational curves via GIT, Osaka J. Math. 48 (2011), no. 4, 1115–1140. MR 2871297
János Kollár, Projectivity of complete moduli, J. Differential Geom. 32 (1990), no. 1, 235–268. MR 1064874
Finn 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, DOI 10.7146/math.scand.a-12001
H.-B. Moon. Birational geometry of moduli spaces of curves of genus zero. Ph.D. dissertation, Seoul National University, 2011.
I. Morrison and D. Swinarski. The $S_{n}$-module structure of $\mathrm {Pic}(\overline {M}_{0,n})$, preprint.
David Mumford, Stability of projective varieties, Enseign. Math. (2) 23 (1977), no. 1-2, 39–110. MR 450272
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
William F. 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, DOI 10.1090/S0002-9947-06-03851-7
M. Simpson. On log canonical models of the moduli space of stable pointed curves. arXiv:0709.4037.