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Some non-finitely generated Cox rings

Part of: Birational geometry Special varieties Curves

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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