b2 schubert singularity

i've been thinking a bit more about the basepoint singularity of the 2-dimensional schubert variety on the projective light cone of a pseudo-euclidean space of signature (2,3), and there seem to be some pretty nice and retrospectively obvious ways of visualizing it that i hadn't noticed yet the last time that i tried to describe it.

one idea is this: the projective light cone of a pseudo-euclidean vector space of signature (t,s) itself carries a conformal structure of signature (t-1,s-1). (perhaps the most familiar example is t=1, s=3, with the projective light cone being our sky, whose conformal structure is revealed whenever we travel near the speed of light.) as i discussed somewhere here recently, a conformal structure is a field or "distribution" of homogeneous quadratic subspaces (called "light cones") of tangent spaces. and the light cone in the tangent space of a point p of the projective light cone of the (2,3) pseudo-euclidean vector space is the tangent cone of the 2-dimensional schubert variety built at the basepoint p. so it's a nice conical singularity, about as i was imagining. the schubert variety is "the 2-dimensional light beams emanating from a 1-dimensional light beam in the (2,3) signature geometry" (so to speak...), which infinitesimally projectively comes down to "the 1-dimensional light beams emanating from a 0-dimensional light beam in the (1,2) signature geometry", which is essentially just the light cone at the point of the projective light cone. (or something like that.)