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Examples of Non-finitely Generated Cox Rings

Part of: Special varieties Birational geometry Cycles and subschemes

Published online by Cambridge University Press:  09 January 2019

José Luis González  and
Kalle Karu
José Luis González
Affiliation:
Department of Mathematics, University of California, Riverside, Riverside, CA 92521, USA Email: jose.gonzalez@ucr.edu
Kalle Karu
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2 Email: karu@math.ubc.ca
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Abstract

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We bring examples of toric varieties blown up at a point in the torus that do not have finitely generated Cox rings. These examples are generalizations of our earlier work, where toric surfaces of Picard number 1 were studied. In this article we consider toric varieties of higher Picard number and higher dimension. In particular, we bring examples of weighted projective 3-spaces blown up at a point that do not have finitely generated Cox rings.

Keywords

Cox ringMori dream spacetoric variety

MSC classification

Primary: 14M25: Toric varieties, Newton polyhedra
Secondary: 14C20: Divisors, linear systems, invertible sheaves 14E30: Minimal model program (Mori theory, extremal rays)

Information

Type
Article
Information
Canadian Mathematical Bulletin , Volume 62 , Issue 2 , June 2019 , pp. 267 - 285
Copyright
© Canadian Mathematical Society 2018 

Footnotes

The first author was supported by the UCR Academic Senate. The second author was supported by a NSERC Discovery grant.

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