Enumerative geometry was originally concerned with determining the number of real objects satisfying independent conditions. Geometers swiftly realized that complex geometry provides the most elegant setting for the resolution of such questions. We extend the classical Schubert calculus of enumerative geometry for the Grassmannian of lines in projective space from the complex realm back to the real.
The primary tool we use in establishing our results
is an adaptation of the `principle of degeneration' from classical
enumerative geometry.
Classical geometers solved enumerative problems by specializing
the conditions so that the number of
figures satisfying the degenerate conditions could be readily
determined.
Then they argued that this number was
indeed the correct number for generic conditions.
Enumerative problems are not generally amenable to a direct
application of this principle; geometers typically use it to determine
the Chow ring, and then solve enumerative problems using the Chow ring.