  We present a result which can be used for stratifications with conical
singularities to deduce that a perverse sheaf (in particular, an intersection
homology sheaf) has reducible characteristic variety, given a hypothesis on the
monodromy of the vanishing cycles local system of a stratum. We apply it to
explain most of the examples currently known where SS(IC(X)) is reducible for X
a Schubert variety in a flag variety.
