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Shapiro and Shapiro's original conjecture

Here is an excerpt from a letter in which Boris Shapiro reminded Sottile of the conjectures he communicated the previous May.
Letter from Boris Shapiro in the Autumn of 1995.

....I want to remind you of my explicit conjecture how to find fully real arrangements of Schubert cells of a given type at least in the space of complete flags. (But apparently this holds also for incomplete flags as well.) I had it in mind and formulated it to you last Spring.

Consider a rational normal curve \gamma in real projective space. Choose any number of osculating complete flags f1, ..., fk to \gamma. Take in the space of complete flags k Schubert cell deompositions w.r.t. eack of the above flags and pick one cell from each of these decompositions.

Conjecture (about 1993). The intersection of these cells is fully real (if nonempty), i.e. the sum of its Z2-Betti numbers coincides with the sum of Betti numbers of its complexification. (In particular, if the sum of codimensions of Schubert cells equals n(n - 1)/2, then all points in this 0-dimensional intersection are real.)


This is similar to an earlier conjecture of Boris Shapiro concerning the phenomenon of reality in enumerative geometry. Here is an email in which he formulates that conjecture.

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