Further remarks

5.ii. Further remarks

While the counterexample of the previous section shows that we cannot guarantee all points of intersection real in a general flag variety when the Schubert varieties are given by flags osculating a real rational normal curve, a number of questions remain (besides the resolution of the conjectures of the previous sections). There remains the original question of Fulton.

Question 1: Given Schubert data for a flag manifold, do there exist real flags in general position whose corresponding Schubert varieties have only real points of intersection?

In every case we know, this does happen. For instance, in the counterexample of the previous section, if the 3-plane B osculates the rational normal curve at 2 (and the others remain fixed), then all 4 solution flags are real. There is also the following result, showing this holds in infinitely many cases. A Grassmannian Schubert condition is a Schubert condition on flags which only imposes conditions on one of the subspaces. We likewise define Grassmannian Schubert data. For example, the Counterexample involves Grassmannian Schubert data. Let F(2,n-2;n) be the manifold of two-step flags consisting of a 2-planes X contained in a (n-2)-plane Y in Cⁿ.

Proposition 1. (Theorem 13 of [So97b]) Given any Grassmannian Schubert data for F(2,n-2;n), there exist real flags whose corresponding Schubert varieties meet transversally with all points of intersection real.

The beauty of the conjectures of Shapiro and Shapiro is that they give a simple algorithm for selecting the flags defining the Schubert varieties.

Question 2: Can the choice of flags in Question 1 (or even Proposition 1) be made effective? In particular, is there an algorithm for selecting these flags?

Also, the cases we were able to prove of Shapiros' conjecture relied upon our ability to show that certain discriminants were sums of squares. Is this always the case?

Question 3: Are the discriminants of polynomial systems arising from Conjectures 1 and 2 always sums of squares?

We only used that these were sums of squares to show that there were no points where the corresponding polynomial systems had multiple solutions. Is this always the case?

Question 4: Do the choices of flags defining Schubert conditions in these conjectures of Shapiro (osculating at distinct real points for Conjectures 1 and 2 always give non-degenerate systems with no multiple solutions (points of intersection)?