The set of k-dimensional subspaces (k-planes) of an n-dimensional vector space forms the Grassmannian of k-planes in n-space, Gr(k, n), which is a smooth projective variety of dimension k(n - k). Those k-planes meeting a linear subspace L of dimension n-k+1-l non-trivially (that is, the intersection has positive dimension) form the special Schubert subvariety X(L) of Gr(k, n) which has codimension l. The special Schubert calculus is concerned with the following question.
|K meets Li non-trivially, for each i=1, 2, ..., s ?||(4.2)|
The condition (4.2) is expressed in the global geometry of Gr(k, n) as the number of points in the intersection of the special Schubert varieties
|X(L1), X(L2), ..., X(Ls),||(4.3)|
There are algorithms due to
Schubert [Sch4] (when each
li is 1) and
Pieri [Pi] to compute the
expected number of solutions.
When each li is 1,
Schubert [Sch2] showed that
the number of solutions is equal to
A line in Pn is a 2-plane in (n+1)-space and two linear subspaces in Pn meet if and only if the corresponding linear subspaces in (n+1)-space have a non-trivial intersection. Thus the problem of lines in projective space corresponds to the case k=2 of the special Schubert calculus. While the geometric problem generalizes easily from k=2 to arbitrary values of k, the proof of Theorem 4.2 does not. There is, however, a relatively simple argument that this special Schubert calculus is fully real.
We present an elementary proof of this result in the important special case
when each li equals 1 so that the conditions are simple,
meaning each Special Schubert variety X(Li)
is a hypersurface in the Grassmannian (has codimension 1).
This proof generalizes to show that some other classes of enumerative problems
in the Schubert calculus are fully real (see
Sections 4.iii.b, 4.iii.c, and 4.iii.d ).
This generalization constructs sufficiently many real solutions using a
limiting argument, as in
Just as the arguments of
Section 2.iii.a. were linked to the homotopy
algorithms of Huber and Sturmfels, the proof of
leads to numerical homotopy methods for solving these