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Another conjecture of Shapiro

... I was interested in the cases when intersection of ANY arrangement of Schubert cells has the M-property. You are interested in the cases when there EXISTS some open set of arrangements of real Schubert cells which intersects in all real points. I want to adjust my conjecture to your case.

Let S₁,..., S_k be some arrangement of real Schubert cells such that S₁\cap...\cap S_k is 0-dimensional and consists of the necessary number of points (counted with multiplicities).

Conjecture. Intersection of any SUBSET of the above cells has the M-property, i.e. the sum of Betti numbers is the same as for the complexified arrangement.

If your question is open for arrangements of Schubert cells in the space of complete flags let me know. I may have a wild guess how one constructs such arrangements (but this is only a wild guess at the moment).


B.Sh.