The first non-trivial instance of the Schubert calculus is the question posed at the beginning of Section 3.i.

Three pairwise skew lines *l*_{1}, *l*_{2}, and
*l*_{3} lie on a unique
smooth quadric surface *Q*.
There are two families of lines that foliate *Q*--one family includes
*l*_{1}, *l*_{2}, and
*l*_{3} and the other consists of the lines meeting each
of *l*_{1}, *l*_{2}, and
*l*_{3}.
The fourth line *l*_{4} meets *Q* in two points, and each of
these points determines a line in the second family.
Thus there are 2 lines *m*_{1} and *m*_{2} in
space that meet general lines
*l*_{1}, *l*_{2}, *l*_{3},
and *l*_{4}.
Figure 10 shows this configuration.

Figure 10:
The two lines meeting four general lines in space. |

The classical Schubert calculus is a vast generalization of this problem of four lines. In the 1980's Speiser suggested to Fulton that the classical Schubert calculus may be a good testing ground for Question 3.1. This was also considered by Chiavacci and Escamilla-Castillo [CE-C]. We will discuss increasingly more general versions of the Schubert calculus, and the status of Question 3.1 for each.

Consider first more general problems involving lines.
The space of lines in **P**^{n} is a smooth projective
variety of dimension 2*n* - 2 called the Grassmannian of lines in
**P**^{n}.
The set of lines meeting a linear subspace *L* of dimension
*n*-1-*l* has
codimension *l* in the Grassmannian.
Thus given general linear subspaces
*L*_{1}, *L*_{2}, ..., *L _{s}*,
of

(4.1) |

These enumerative problems of lines in **P**^{n}
meeting general linear subspaces furnished the first infinite family of
non-trivial enumerative problems known to be fully real.