Ueber Hilberts vierzehntes Problem fuer Varietaeten mit Kompliziertheit eins

Let G be a connected reductive group with Borel subgroup B acting on a normal variety X. The complexity of X is lowest codimension of a B-orbit.

Theorem: If X is unirational (e.g. quasi-homogeneous) of complexity at most one then the ring k[X] of global functions is finitely generated.

This result is sharp: If X is not unirational or of complexity at least two then k[X] might not be finitely generated.

Appeared in: Mathematische Zeitschrift 213 (1993) 33-35

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Remark: For an application of the theorem see my paper with Michel Brion.

Last updated: March 10, 2004