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Hilbert's 14th problem and Cox rings

Published online by Cambridge University Press:  24 November 2006

Ana-Maria Castravet  and
Jenia Tevelev
Ana-Maria Castravet
Affiliation:
Department of Mathematics, University of Texas at Austin, Austin, TX 78712, USA Department of Mathematics, University of Massachusetts, Amherst, MA 01003-9305, USA. noni@math.umass.edu
Jenia Tevelev
Affiliation:
Department of Mathematics, University of Texas at Austin, Austin, TX 78712, USA Department of Mathematics, University of Massachusetts, Amherst, MA 01003-9305, USA. tevelev@math.umass.edu
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Abstract

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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.

Keywords

Hilbert's 14th problemCox ringsMori dream spacesrings of invariants14E14M
Type
Research Article
Information
Compositio Mathematica , Volume 142 , Issue 6 , November 2006 , pp. 1479 - 1498
Copyright
Foundation Compositio Mathematica 2006