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Some non-finitely generated Cox rings

Part of: Curves Special varieties Birational geometry

Published online by Cambridge University Press:  22 December 2015

José Luis González  and
Kalle Karu
José Luis González
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada email [email protected] Current address:Department of Mathematics, Yale University, New Haven, CT 06511, USA
Kalle Karu
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada email [email protected]
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Abstract

We give a large family of weighted projective planes, blown up at a smooth point, that do not have finitely generated Cox rings. We then use the method of Castravet and Tevelev to prove that the moduli space $\overline{M}_{0,n}$ of stable $n$-pointed genus-zero curves does not have a finitely generated Cox ring if $n$ is at least $13$.

Keywords

Cox ringsmoduli spaces of curvesMori dream spacesweighted projective planes

MSC classification

Primary: 14M25: Toric varieties, Newton polyhedra 14E30: Minimal model program (Mori theory, extremal rays) 14H10: Families, moduli (algebraic)
Type
Research Article
Information
Compositio Mathematica , Volume 152 , Issue 5 , May 2016 , pp. 984 - 996
Copyright
© The Authors 2015 

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References

Birkar, C., Cascini, P., Hacon, C. and McKernan, J., Existence of minimal models for varieties of log general type, J. Amer. Math. Soc. 23 (2010), 405–468.Google Scholar
Castravet, A.-M., The Cox ring of M0, 6, Trans. Amer. Math. Soc. 361 (2009), 3851–3878.Google Scholar
Castravet, A.-M. and Tevelev, J., M0, n is not a Mori dream space, Duke Math. J. 164 (2015), 1641–1667; MR 3352043.CrossRefGoogle Scholar
Cutkosky, S. D., Symbolic algebras of monomial primes, J. Reine Angew. Math. 416 (1991), 71–89; MR 1099946 (92b:13009).Google Scholar
Fulton, W., Introduction to toric varieties, Annals of Mathematics Studies, vol. 131 (Princeton University Press, Princeton, NJ, 1993); MR 1234037 (94g:14028).Google Scholar
Goto, S., Nishida, K. and Watanabe, K., Non-Cohen–Macaulay symbolic blow-ups for space monomial curves and counterexamples to Cowsik’s question, Proc. Amer. Math. Soc. 120 (1994), 383–392; MR 1163334 (94d:13005).Google Scholar
Hu., Y. and Keel, S., 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 (2001i:14059).CrossRefGoogle Scholar
Okawa, S., On images of Mori dream spaces, Math. Ann. (2015), doi:10.1007/s00208-015-1245-5.Google Scholar