SAGBI bases and Degeneration of Spherical Varieties to Toric Varieties

Details SAGBI bases and Degeneration of Spherical Varieties to Toric Varieties SAGBI bases and Degeneration of Spherical Varieties to Toric Varieties

2003-09-25

arxiv.org/abs/math/0309413v1

Mathematics

Algebraic Geometry

17 pages, LaTex, uses the package xy

Scientific paper

Let X \subset Proj(V) be a projective spherical G-variety, where V is a finite dimensional G-module and G = SP(2n, C). In this paper, we show that X can be deformed, by a flat deformation, to the toric variety corresponding to a convex polytope \Delta(X). The polytope \Delta(X) is the polytope fibred over the moment polytope of X with the Gelfand-Cetlin polytopes as fibres. We prove this by showing that if X is a horospherical variety, e.g. flag varieties and Grassmanians, the homogeneous coordinate ring of X can be embedded in a Laurent polynomial algebra and has a SAGBI basis with respect to a natural term order. Moreover, we show that the semi-group of initial terms, after a linear change of variables, is the semi-group of integral points in the cone over the polytope \Delta(X). The results of this paper are true for other classical groups, provided that a result of A. Okounkov on the representation theory of SP(2n,C) is shown to hold for other classical groups.

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