schubert varieties of b2

as suggested earlier, i want to use the 2-dimensional schubert varieties in the grassmanians of b2 to test some ideas about when schubert varieties are non-singular.

we can think of these grassmanians as the spaces of partial linear isometries of ranks 1 and 2, respectively, from euclidean 2-space to euclidean 3-space. (thus the rank 2 ones are actually total linear isometries.) these are 3-dimensional projective varieties.

the 2-dimensional schubert variety in the grassmanian of rank 1 plis consists of all those that are compatible with a fixed one of them, in the sense that both are restrictions of some tli.

the 2-dimensional schubert variety in the grassmanian of tlis consists of all those whose equalizer with a fixed one of them is at least rank 1.

my naive prediction seems to be that the first of these schubert varieties will be non-singular and the second will be singular, because there's an invariant 2-dimensional distribution on the first grassmanian but not on the second one.

let me try to argue that the first half of the prediction is correct and that in fact the first schubert variety is a projective plane, isomorphic to the projective space of the 3-dimensional target space. given a 1-dimensional linear subspace v of the target space, there's a unique pli from the source space with image v such that .... hmmm, this doesn't seem to be working.... ??

worse yet, the other half of the prediction seems to be failing too. naturally i suspect that i've got something backwards somewhere, but i've been looking for such a mistake and haven't found it yet.

i suppose i should consider the possibility that the problem involves using my intuition about real-algebraic geometry in a situation where maybe complex-algebraic would be more appropriate. i doubt that that's the problem though.

ok, wait, i think that i found the problem, and that i've almost got this straightened out now. i seem to have made a pretty silly mistake, but i should take a bit of time to make sure that i've really got it straightened out now.

ok, so the mistake is that i got it backwards when i said:

my naive prediction seems to be that the first of these schubert varieties will be non-singular and the second will be singular, because there's an invariant 2-dimensional distribution on the first grassmanian but not on the second one.

instead it should have been:

the first of these schubert varieties will be singular and the second will be non-singular, because there's an invariant 2-dimensional distribution on the second grassmanian but not on the first one.

in trying to straighten out which one has the invariant 2-dimensional distribution, i found it helpful to remind myself about the other way of thinking about b2, associating it with the geometry of a 4-dimensional symplectic vector space instead of that of a (2+3)-dimensional pseudo-euclidean vector space. then the first grassmanian is the "lagrangian grassmanian" of 2-dimensional isotropic subspaces while the second grassmanian is that of 1-dimensional isotropic subspaces. this latter is just the projective space of the symplectic vector space, which carries a natural contact distribution which is the invariant 2-dimensional distribution that we're supposed to be thinking about here.

if we really do have this straightened out now then i'd like to go further and try to understand the nature of the basepoint singularity of the 2-dimensional schubert variety in the first grassmanian in greater detail, for example what the tangent cone is like; i'm still confused about some aspects of that.