Skip to main content

Examples of Non-finitely Generated Cox Rings

  • José Luis González (a1) and Kalle Karu (a2)

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.

Hide All

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

Hide All
[1] Berchtold, F. and Hausen, J., Cox rings and combinatorics . Trans. Amer. Math. Soc. 359(2007), no. 3, 12051252.
[2] Castravet, A.-M., Mori dream spaces and blow-ups. In: Algebraic Geometry: Salt Lake City 2015. Proceedings of Symposia in Pure Mathematics, 97(1), American Mathematical Society, Providence, RI, 2018, pp. 143–168.
[3] Castravet, A.-M. and Tevelev, J., M̄0, n is not a Mori dream space . Duke Math. J. 164(2015), 16411667.
[4] Cutkosky, S. D., Symbolic algebras of monomial primes . J. Reine Angew. Math. 416(1991), 7189.
[5] Fulton, W., Introduction to toric varieties. Annals of Mathematics Studies, 131, Princeton University Press, Princeton, NJ, 1993.
[6] González, J. L. and Karu, K., Some non-finitely generated Cox rings . Compos. Math. 152(2016), 984996.
[7] 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), 383392.
[8] He, Z., New examples and non-examples of Mori dream spaces when blowing up toric surfaces. 2017. arxiv:1703.00819.
[9] Hu, Y. and Keel, S., Mori dream spaces and GIT . Michigan Math. J. 48(2000), 331348.
[10] Okawa, S., On images of Mori dream spaces . Math. Ann. 364(2016), 13151342.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Canadian Mathematical Bulletin
  • ISSN: 0008-4395
  • EISSN: 1496-4287
  • URL: /core/journals/canadian-mathematical-bulletin
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


MSC classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed