4.i. The Schubert Calculus of Lines

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

Three pairwise skew lines l₁, l₂, and l₃ lie on a unique smooth quadric surface Q. There are two families of lines that foliate Q--one family includes l₁, l₂, and l₃ and the other consists of the lines meeting each of l₁, l₂, and l₃. The fourth line l₄ meets Q in two points, and each of these points determines a line in the second family. Thus there are 2 lines m₁ and m₂ in space that meet general lines l₁, l₂, l₃, and l₄. 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 Pⁿ is a smooth projective variety of dimension 2n - 2 called the Grassmannian of lines in Pⁿ. 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 L₁, L₂, ..., L_s, of Pⁿ with the dimension of L_i equal to n-1-l_i, where we have l₁ + l₂ + ... + l_s = 2n - 2, we expect (and there are indeed) finitely many lines in Pⁿ meeting each linear subspace L₁, L₂, ..., L_s. Schubert [Sch3] discovered algorithms for computing this number d(l₁,l₂,...,L_s) of lines. For example, if each l_i is 1, so that s = 2n - 2, then this number is the nth Catalan number^§

(4.1)

These enumerative problems of lines in Pⁿ meeting general linear subspaces furnished the first infinite family of non-trivial enumerative problems known to be fully real.