## Contractions and flips for varieties with group action of small complexity

J. Math. Sci. Univ. Tokyo
Vol. 1 (1994), No. 3, Page 641--655.

Brion, Michel ; Knop, Friedrich
Contractions and flips for varieties with group action of small complexity

Abstract:
We consider projective, normal algebraic varieties $X$ equipped with the action of a reductive algebraic group $G$. We assume that a Borel subgroup of $G$ has an orbit of codimension at most one in $X$ (i.e.\ the complexity of the $G$-variety $X$ is at most one) and that $X$ is unirational. Then we prove that the cone of effective one-cycles $NE(X)$ is finitely generated, and that each face of $NE(X)$ can be contracted. Moreover, flips exist when $X$ is $\bold Q$-factorial, and any sequence of directed flips terminates. Finally, we prove that any homogeneous space of complexity at most one admits an equivariant completion whose anticanonical divisor is ample.

Mathematics Subject Classification (1991): Primary 14L30; Secondary 14E30, 14M17
Mathematical Reviews Number: MR1322696