  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of G and T a maximal torus in B. A beautiful unpublished result of Dale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.
