Shapiro Conjecture : Grassmannian Conditions on Flags

Grassmannian Conditions on Flags

A partial flag E. of type b is a sequence of subspaces E_b₁, E_b₂, ..., E_{b_k}, where E_{b_i} has dimension b_i, and each subspace is contained in the next.

A Grassmannian condition of index b_i on a flag E. of type b is a general incidence condition imposed on the subspace E_{b_i} by (part of) some (full) flag F.

Example
We could consider flags of type (1,2) in 3-space. These are lines m lying on planes H.
Requiring m to meet a fixed line is a Grassmannian condition of index 1 on (m,H).
Requiring H to contain a fixed point is a Grassmannian condition of index 2.

Consider flags of type (1,2) in 3-space satisfying
three of these conditions of index 1 and two of these conditions of index 2.

Since H meets two points, it contains the line l they span, and so m also meets l.
If the points and lines osculate the rational normal curve, then this is the problem of tangent and secant lines we saw.