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I'm interested in what is known about the stratification of the Grassmanian induced by the coordinate hyperplanes in the Plucker embedding. More specifically, if we view the projective space as a toric variety with boundary the coordinate hyperplanes, what is the intersection of a torus orbit (and not simply an orbit closure) with the Grassmanian?
I'm particularly interested in the following questions: 1. Is the intersection of a codim k orbit codim k on the Grassmanian? 2. Is each intersection irreducible? 3. When is it smooth?

I doubt any of this is known or even true in general, but I'd even be interested in the cases of G(2,n) and G(3,n).

Thanks!

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    $\begingroup$ This is the same as the matroid stratification of the Grassmannian (since each Plucker coordinate tells you whether certain rows are dependent or not). Much is known, but Mnev's theorem also says that arbitrarily nasty things occur. $\endgroup$
    – Alexander Woo
    Jul 18, 2015 at 21:42
  • $\begingroup$ Thanks. I did some searching on Mnev's theorem, and see that 3 is certainly hopeless. However, I don't fully understand the theorem since I know nothing about real algebraic geometry. So, when we say arbitrarily nasty things occur, does this include having multiple components of various dimensions? $\endgroup$
    – user71216
    Jul 19, 2015 at 14:52
  • $\begingroup$ Sorry - I've told you pretty much everything I know. You might search using 'matroid stratification' also - or hope someone more knowledgeable comes along! $\endgroup$
    – Alexander Woo
    Jul 20, 2015 at 2:35

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