a2 schubert varieties

i've thought about the schubert varieties of a2 a bit before, in connection with invariant contact distributions on flag varieties. the a2 flag variety can be thought of as the projective cotangent bundle of the projective plane, which is close to the conceptually most natural sort of contact manifold; you can picture its points as infinitesimal little slats of a venetian blind, with the contact distribution corresponding to the blind being closed rather than open. (it's odd though that the non-orientability of the projective plane prevents the blind from providing much shade.) the big bruhat cell corresponds to the 3-dimensional heisenberg lie group which can also be thought of as the "jet bundle" of first-order taylor serieses of functions on the line, and which embeds into the projective cotangent bundle of the projective plane in a fairly obvious way. a first-order non-linear time-independent partial differential equation on the line lives as a vector field on this jet bundle, and if a wave "breaks" in finite time due to the non-linearity of the wave equation then its evolution may be continued outside the image of the embedding.

now however i want to re-examine the a2 schubert varieties from the viewpoint of my recent focus on singularities of schubert varieties. it might turn out that these particular schubert varieties lack singularities but that could still be interesting by way of contrast.

(on the other hand might relationships between singularities and "caustics" be lurking here somewhere? this seems like a long shot, offhand.)