On the Smooth Points of T-stable Varieties in G/B and the Peterson Map

Details On the Smooth Points of T-stable Varieties in G/B and the Peterson Map On the Smooth Points of T-stable Varieties in G/B and the Peterson Map

2000-05-02

arxiv.org/abs/math/0005025v1

Invent. Math. Online First November 8, 2002

Mathematics

Algebraic Geometry

Scientific paper

Let G be a semi-simple algebraic group over ${\mathbb C}$, B a Borel subgroup of G and T a maximal torus in B. A beautiful unpublished result of Dale Peterson says that if G is simply laced, then every rationally smooth point of a Schubert variety X in G/B is nonsingular in X. The purpose of this paper is to generalize this result to arbitrary T-stable subvarieties of G/B, the only restriction being that G contains no $G_2$ factors. In particular, we show that a Schubert variety X in such a G/B is nonsingular if and only if all the reduced tangent cones of X are linear.

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