Explicit Enumerative Geometry for the Real Grassmannian of Lines in Projective Space

For any collection of Schubert conditions on lines in projective space which generically determine a finite number of lines, we show there exist real generic conditions determining the expected number of real lines. This extends the classical Schubert calculus of enumerative geometry for the Grassmann variety of lines in projective space from the complex realm to the real. Our main tool is an explicit description of rational equivalences which also constitutes a novel determination of the Chow rings of these Grassmann varieties. The combinatorics of these rational equivalences suggests a non-commutative associative product on the free abelian group on Young tableaux. We conclude by considering some enumerative problems over other fields.