Spherical varieties and Mori theory
This article contains many interesting results on \(B\)-stable curves on normal \(G\)-varieties. For example: In a projective spherical variety each irreducible \(B\)-stable curve is isomorphic to the projective line. Here \(G\) is a connected reductive algebraic group (over an algebraically closed field of characteristic 0) and \(B\) is a Borel subgroup of \(G\). A normal algebraic variety with \(G\)-action is spherical if \(B\) contains a dense orbit. A spherical variety is uniquely determined by its ``coloured fan'' (certain combinatorial data). The main objective of this paper is to study spherical varieties from the viewpoint of Mori theory. [Thus results of \textit{M. Reid} in: Arithmetic and geometry, Vol. II, Prog. Math. 36, 395-418 (1983; Zbl 0571.14020) for toric varieties are generalized.] For a complete spherical variety \(X\) the author proves the following: \(\text{NE}(X)\), the cone of effective 1-cycles, is polyhedral and generated by the classes of \(B\)-stable rational curves. A system of generators is described in terms of the coloured fan of \(X\). Moreover, flips in spherical \(\mathbb{Q}\)-factorial varieties are studied.
scientific article; zbMATH DE number 558967
scientific article