The Cox ring of

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 Free Access

Abstract | References | Similar Articles | Additional Information

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

**[BCHM]**Birkar, C., Cascini, P., Hacon, C., McKernan, J.,*Existence of minimal models for varieties of log general type*, preprint (2006)**[BP]**Victor V. Batyrev and Oleg N. Popov,*The Cox ring of a del Pezzo surface*, Arithmetic of higher-dimensional algebraic varieties (Palo Alto, CA, 2002), Progr. Math., vol. 226, Birkhäuser Boston, Boston, MA, 2004, pp. 85–103. MR**2029863**, 10.1007/978-0-8176-8170-8_5**[CT]**Ana-Maria Castravet and Jenia Tevelev,*Hilbert’s 14th problem and Cox rings*, Compos. Math.**142**(2006), no. 6, 1479–1498. MR**2278756**, 10.1112/S0010437X06002284**[EH]**David Eisenbud and Joe Harris,*The geometry of schemes*, Graduate Texts in Mathematics, vol. 197, Springer-Verlag, New York, 2000. MR**1730819****[GKM]**Angela Gibney, Sean Keel, and Ian Morrison,*Towards the ample cone of \overline𝑀_{𝑔,𝑛}*, J. Amer. Math. Soc.**15**(2002), no. 2, 273–294. MR**1887636**, 10.1090/S0894-0347-01-00384-8**[HT]**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**, 10.1090/conm/314/05424**[HK]**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**, 10.1307/mmj/1030132722**[KM]**Keel, S., McKernan, J.,*Contractible extremal rays of*, preprint (1997), arxiv:alg-geom/9707016**[V]**Peter Vermeire,*A counterexample to Fulton’s conjecture on \overline𝑀_{0,𝑛}*, J. Algebra**248**(2002), no. 2, 780–784. MR**1882122**, 10.1006/jabr.2001.9044

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

Email:
noni@math.umass.edu, noni@math.arizona.edu

DOI:
https://doi.org/10.1090/S0002-9947-09-04641-8

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.