Singularities of Schubert varieties, tangent cones and Bruhat graphs

Let G be a semi-simple algebraic group without G₂-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 Θx(X) = dim X, where Θx(X) is the linear span of the reduced tangent cone to X at x. The purpose of this paper is to describe Θx(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, Θx(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 B₂-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.