Schubert problems with respect to osculating flags of stable rational curves
Abstract
Given a point z in P1, let F(z) be the osculating flag to the rational normal curve at point z. The study of Schubert problems with respect to such flags F(z1), F(z2), ..., F(zr) has been studied both classically and recently, especially when the points zi are real. Since the rational normal curve has an action of PGL2, it is natural to consider the points (z1, ..., zr) as living in the moduli space of r distinct point in P1 -- the famous M0,r. One can then ask to extend the results on Schubert intersections to the compactification M0,r. The first part of this paper achieves this goal. We construct a flat, Cohen-Macaulay family over M0,r, whose fibers over M0,r are isomorphic to G(d,n) and, given partitions lambda1, ..., lambdar, we construct a flat Cohen-Macualay family over M0,r whose fiber over (z1, ..., zr) in M0,r is the intersection of the Schubert varieties indexed by lambdai with respect to the osculating flags F(zi). In the second part of the paper, we investigate the topology of the real points of our family, in the case that sum |lambdai| = dim G(d,n). We show that our family is a finite covering space of M0,r, and give an explicit CW decomposition of this cover whose faces are indexed by objects from the theory of Young tableaux.