Singularities of Schubert varieties, tangent cones and Bruhat graphs

Abstract

Let G be a semi-simple algebraic group without G 2 -factors over an algebraically closed field k of characteristic p ≠ 2, 3, and suppose B is a Borel subgroup, T ⊂ B is a maximal torus, and P is a parabolic in G containing B . In an earlier paper, the authors classified the singular T -fixed points x of an arbitrary irreducible T -stable subvariety X in G/P in all characteristics, the key to this being the notion of a Peterson translate. In particular, we showed that if X is Cohen-Macaulay, then X is smooth at x if and only if there exists a T -invariant curve in X through x which contains a smooth point of X and dim Θ x(X) = dim X , where Θ x(X) is the linear span of the reduced tangent cone to X at x . The purpose of this paper is to describe Θ x(X) when X is a Schubert variety in G/P and x is a maximal singular T -fixed point of X . In fact, we give two characterizations. We first show that in all characteristics, Θ x(X) is the sum of all the Peterson translates at x . The second characterization involves further study of the Peterson translates, along the good T -invariant curves at x , for which the assumption char( k ) ≠ 2, 3 is needed. This leads to the following consequence: if x is a maximal singularity of X which is rationally smooth, then either the span of the tangent lines to the T -stable curves is not a module for the isotropy subgroup of B at x , or there exist a pair of orthogonal T -invariant curves at x which determine what we call a B 2 -pair. This characterization gives a nonrecursive algorithm for finding the singular locus of an arbitrary Schubert variety in G/P in terms of its Bruhat graph.