Lower bounds in the Schubert calculus for flags?

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6.ii Lower bounds in the Schubert calculus for flags?

Conjecture 5.12 speculates that a zero-dimensional intersection of Grassmannian Schubert varieties in a flag manifold Fl_b has only real points, when the Schubert varieties are given by flags osculating the rational normal curve at points whose associated partition is ordered. If the associated partition is not ordered, then we conjectured (and have found experimentally) that there is a selection of osculating flags with all points of intersection real, as well as a selection with not all points real.

The table below shows the results of computing 160,000 instances of the intersection (5.11), recording the number having a given number of real points, and sorting this by the simple Schubert data. The data is a sequence of 4 2's and 4 3's, up to reversal and cyclic permutations, there are 8 such sequences. Observe that the apparent lower bound on the number of real solutions depends on the sequence, which is a measure of the configuration of the conditions. This reinforces the observation made in Section 3.i concerning the subtle dependence of the number of real solutions on the configuration of the conditions. Other calculations we have done reinforce this observation.

Data	Number of Real Solutions
	0	2	4	6	8	10	12
22223333	0	0	0	0	0	0	20000
22332233	0	0	136	1533	7045	5261	6025
22233233	0	0	67	3015	6683	4822	5413
22322333	0	0	9	1677	3835	6247	8232
22323323	0	0	303	2090	6014	7690	3903
22232333	0	195	1476	1776	3628	4546	8379
22323233	0	37	1944	4367	6160	4634	2858
23232323	251	929	5740	3168	5420	2828	1662

Next: 7. Acknowledgements
Up: 6. Lower bounds in the Schubert calculus
Previous: 6.i. The Real Degree of Grassmann Varieties