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## 4.ii. The Special Schubert Calculus

More generally, we may ask how many linear
subspaces of a fixed dimension meet general linear subspaces
*L*_{1}, *L*_{2}, ..., *L*_{s}.
We formulate this question in terms of linear subspaces of a vector
space.
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.

**Question 4.3**
Given general linear subspaces

*L*_{1},

*L*_{2}, ...,

*L*_{s}
of

**C**^{n} where the dimension of

*L*_{i}
is

*n*-

*k*+1-

*l*_{i} and we have

*l*_{1} +

*l*_{2} + ... +

*l*_{s}
equal to

*k*(

*n* -

*k*), how many

*k*-planes

*K*
meet each subspace

*L*_{i} non-trivially, that is, satisfy

*K* meets *L*_{i} 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*(*L*_{1}), *X*(*L*_{2}),
..., *X*(*L*_{s}), |
(4.3) |

when the intersection is transverse.
(A general theorem of Kleiman [Kl1] guarantees transversality when
the *L*_{i} are in general position, and also implies
transversality for the other intersections considered in this section.)
There are algorithms due to
Schubert [Sch4] (when each
*l*_{i} is 1) and
Pieri [Pi] to compute the
expected number of solutions.
When each *l*_{i} is 1,
Schubert [Sch2] showed that
the number of solutions is equal to

A line in **P**^{n} is a 2-plane in
(*n*+1)-space
and two linear subspaces in **P**^{n} 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.

**Theorem 4.4**
([

So5 Theorem 1)
Suppose

*n*,

*k*, and

*l*_{1},

*l*_{2}, ...,

*l*_{s}
are positive integers with

*l*_{1} +

*l*_{2} + ... +

*l*_{s}
equal to

*k*(

*n* -

*k*).
Then there are linear subspaces

*L*_{1},

*L*_{2}, ...,

*L*_{s}
of

**R**^{n} in general position with
the dimension of

*L*_{s} equal to

*n*-

*k*+1-

*l*_{i} such that
each of the a priori complex

*k*-planes

*K*
satisfying (

4.2) are in fact real.

We present an elementary proof of this result in the important special case
when each *l*_{i} equals 1 so that the conditions are simple,
meaning each Special Schubert variety *X*(*L*_{i})
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
Section 2.iii.a.
Just as the arguments of
Section 2.iii.a. were linked to the homotopy
algorithms of Huber and Sturmfels, the proof of
Theorem 4.4
leads to numerical homotopy methods for solving these
problems [HSS,HV].

**Subsections**
- 4.ii.a. Geometry of Grassmann Varieties
- 4.ii.b. Degrees of Schubert Varieties
- 4.ii.c. Proof of Theorem 4.4

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