since i'm using the lie group "b2" as a test case for understanding singularity vs non-singularity of schubert varieties, i'll try to describe here a certain mental picture that i have of the b2 incidence geometry (in the split real case) as what i call "hoop geometry".
among the various names of various forms of b2 are "sp(4)" and "so(2,3)". "sp(4)" means the automorphism group of a 4-dimensional symplectic vector space while "so(2,3)" means the automorphism group of a pseudo-euclidean vector space of signature (2,3). one of the things that the symplectic picture is good for is for understanding the invariant distribution on one of the grassmanians as the natural "contact distribution" on the projective space of a symplectic space, while one of the things that the pseudo-euclidean picture is good for is for the "hoop geometry" idea, which i'll try to explain.
perhaps the most famous pseudo-euclidean geometry is so(1,3), the lorentzian geometry of special relativity. this is the geometry of light beams, which we can think of as linear isometries from euclidean 1-dimensional "time" to euclidean 3-dimensional "space". (light moves in straight lines and always at a certain speed aka "the speed of light".) so(2,3) works the same way except that "time" is now a 2-dimensional euclidean space instead of 1-dimensional; thus now we have the geometry of "2-dimensional light beams" aka linear isometries from 2-dimensional euclidean time to 3-dimensional euclidean space. as with so(1,3) we have symmetries of time alone and of space alone but also "boosts" that mix time and space together.
but now picture the unit sphere in 2-dimensional time as a 1-dimensional "hoop" which fits around an equator of the 2-dimensional unit sphere of 3-dimensional space, sort of like the rings of saturn, thought of as a hula hoop around its equatorial waist.
the grassmanian of total linear isometries from time to space is now just the configuration space of these hoop placements. the other b2 grassmanian, of rank 1 partial linear isometries ("partial light beams"), can be pictured as "lines" of configurations where the hoop swivels around an antipodal pair of nails driven through both the hoop and the sphere; or alternatively as the nail placements themselves, around which the hoop can freely swivel.
now let's try using this mental picture to visualize the 2-dimensional schubert varieties in the grassmanians, and check whether they seem singular or non-singular as predicted.
the 2-dimensional schubert variety in what i'm now thinking of as the "point" grassmanian is the hoop placements that can be reached from an initial such placement by a single swivel. it seems visually clear that this is a non-singular variety ...
well, actually it wasn't as visually clear as i thought at first. nevertheless, the situation seems visualizable enough that after mulling it over for a day or so i think i can see that the real points of the variety form a projective plane, at least in the naive topological sense, which suggests that algebraically it's in fact probably just a projective plane.
we can also revert to the symplectic picture, where the point grassmanian is now the 3d projective space of the 4d symplectic vector space. so the 2d schubert variety in the point grassmanian is just a projective plane in a projective 3-space, quite vanilla it seems.
but now let's try to visualize the 2-dimensional schubert variety in the "line" grassmanian...
actually, first i'm going to dispense with visualization... this schubert variety is the space of rank 1 partial linear isometries from 2d euclidean space ("time") to 3d euclidean space ("space") that are compatible with a given one of them. we can map from this schubert variety to the projective space of the 3d space by taking the image of the partial isometry. this map is 2-to-1 except over the image of the given one, where it's only 1-to-1. is this enough to establish that the variety is singular in a naive topological sense?
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