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2 results

  • Some non-finitely generated Cox rings

  • Part of
  • José Luis González, Kalle Karu
  • Journal:
    Compositio Mathematica / Volume 152 / Issue 5 / May 2016
    Published online by Cambridge University Press:
    22 December 2015, pp. 984-996
    Print publication:
    May 2016
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  • 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 $13$.

  • Hilbert's 14th problem and Cox rings

  • Ana-Maria Castravet, Jenia Tevelev
  • Journal:
    Compositio Mathematica / Volume 142 / Issue 6 / November 2006
    Published online by Cambridge University Press:
    24 November 2006, pp. 1479-1498
    Print publication:
    November 2006
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  • Our main result is the description of generators of the total coordinate ring of the blow-up of $\mathbb{P}^n$ in any number of points that lie on a rational normal curve. As a corollary we show that the algebra of invariants of the action of a two-dimensional vector group introduced by Nagata is finitely generated by certain explicit determinants. We also prove the finite generation of the algebras of invariants of actions of vector groups related to T-shaped Dynkin diagrams introduced by Mukai.