Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The Cox ring of $ \overline{M}_{0,6}$

Author: Ana-Maria Castravet
Journal: Trans. Amer. Math. Soc. 361 (2009), 3851-3878
MSC (2000): Primary 14E30, 14H10, 14H51, 14M99
Published electronically: January 28, 2009
MathSciNet review: 2491903
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that the Cox ring of the moduli space $ \overline{M}_{0,6}$, of stable rational curves with $ 6$ marked points, is finitely generated by sections corresponding to the boundary divisors and divisors which are pull-backs of the hyperelliptic locus in $ \overline{M}_3$ via morphisms $ \rho:\overline{M}_{0,6}\rightarrow \overline{M}_3$ that send a $ 6$-pointed rational curve to a curve with $ 3$ nodes by identifying $ 3$ pairs of points. In particular this gives a self-contained proof of Hassett and Tschinkel's result about the effective cone of $ \overline{M}_{0,6}$ being generated by the above mentioned divisors.

References [Enhancements On Off] (What's this?)

  • [BCHM] Birkar, C., Cascini, P., Hacon, C., McKernan, J., Existence of minimal models for varieties of log general type, preprint (2006)
  • [BP] Batyrev, V., Popov, O., The Cox ring of a del Pezzo surface, Arithmetic of higher-dimensional algebraic varieties, Palo Alto, CA, 2002, 149-173, Progr. Math., 226, Birkhäuser Boston, Boston, MA, 2004 MR 2029863 (2005h:14091)
  • [CT] Castravet, A.-M., Tevelev, J., Hilbert's 14'th Problem and Cox Rings, (2005), Compositio Math., Vol. 142 (2006), 1479-1498 MR 2278756 (2007i:14044)
  • [EH] Eisenbud, D., Harris, J., The Geometry of Schemes, Graduate Texts in Mathematics, Vol. 197, Springer-Verlag, New York, 2000 MR 1730819 (2001d:14002)
  • [GKM] Gibney, A., Keel, S., Morrison, I., Towards the ample cone of $ \overline{M}_{g,n}$, J. Amer. Math. Soc., Vol. 15, No. 2 (2001), 273-294 MR 1887636 (2003c:14029)
  • [HT] Hassett, B., Tschinkel, Y., On the effective cone of the moduli space of pointed rational curves, Topology and geometry: commemorating SISTAG, 83-96, Contemp. Math., 314, Amer. Math. Soc., Providence, RI, 2002 MR 1941624 (2004d:14028)
  • [HK] Hu, Y., Keel, S., Mori Dream Spaces and GIT, Michigan Math. J., Vol. 48 (2000), 331-348 MR 1786494 (2001i:14059)
  • [KM] Keel, S., McKernan, J., Contractible extremal rays of $ \overline{M}_{0,n}$, preprint (1997), arxiv:alg-geom/9707016
  • [V] Vermeire, P., A counterexample to Fulton's Conjecture on $ \overline{M}_{0,n}$, J. of Algebra, Vol. 248, (2002), 780-784 MR 1882122 (2002k:14043)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 14E30, 14H10, 14H51, 14M99

Retrieve articles in all journals with MSC (2000): 14E30, 14H10, 14H51, 14M99

Additional Information

Ana-Maria Castravet
Affiliation: Department of Mathematics, University of Massachusetts at Amherst, Amherst, Massachusetts 01003
Address at time of publication: Department of Mathematics, University of Arizona, Tucson, Arizona 85721

Keywords: Cox rings, Mori Dream Spaces, moduli spaces of stable curves
Received by editor(s): May 4, 2007
Received by editor(s) in revised form: September 24, 2007
Published electronically: January 28, 2009
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society