A Discriminant 25
Consider again the lines meeting three tangent and one secant line to the rational normal curve. (The secant points correspond to conditions of index 2 on partial flags of type (1,2).)

    Assume that the lines are tangent at the points corresponding to g(-1), g(0), and g(1), and the two secant points are at g(v) and g(w). We can compute a discriminant for this problem

16 (v - w)2 (2 v w + v + w) (3 v w + 1) (1 - v w) (v + w - 2 v w) .
Note that
1-vw  =  1-w + w(1-v)
v + w - 2 v w  =  v(1-w) + w(1-v)
and so the discriminant is positive when 0 < v,w < 1, which is a monotone choice of the points.

We display this discriminant in the v, w plane.
Yellow is where it is negative.
The dotted lines and the axes are where v or w are equal to one of the tangent points.

Note that it is positive when v and w lie in the same interval between two points of tangency.