A counterexample

5.i. A counterexample to the original conjecture of Shapiro and Shapiro

The original conjecture of Shapiro and Shapiro was quite general and dealt with the M-property for intersections of Schubert cells on a flag manifold. Here is an excerpt from a letter they sent to Sottile describing it. The original conjecture was also too optimistic, however, we believed it up until we found a counterexample.

In the manifold of partial flags in C⁵ consisting of a 2-plane X contained in a 3-plane Y, consider the following enumerative problem: Given general 3-planes A,B, and C and general 2-planes a,b, and c in C⁵, how many such flags are there where

X meets each of a,B, and C non-trivially, and
Y meets b and c non-trivially, and it also intersects A in a 2-plane?

Shapiros' original conjecture in this case is that if A,B,C and a,b,c all osculate a rational normal curve at distinct real points, then all partial flags satisfying these conditions are real.

The calculus of enumerative geometry for the flag manifold (for example, a repeated application of the Pieri-type formula of [So96]) shows there will be 4 partial flags satisfying these conditions. Worth noting about this problem is that it is the simplest enumerative problem in a flag manifold that we cannot reduce to a problem in a Grassmann variety.

In the Summer of 1998, we computed a few instances of this problem, and for these, all 4 solutions were real. Inspired by this, we tried to prove the conjecture in this case by computing the discriminant of an equivalent polynomial system in some local coordinates. While studying the discriminant, we noted that it is not always non-negative, which led us to factor it and look closer at examples where the discriminant was negative. In this way, we found a counterexample to Shapiros' conjecture. In this counterexample, the 2-planes a,b, and c in C⁵ osculate the standard rational normal curve at the points 4, 1, and -5, while the 3-planes A,B, and C osculate at the points 0, -1, and 3. Here is a documented Maple V.5 script for you to examine and test for yourself (The output of that script is here)