Schubert problems with respect to osculating flags of stable rational curves

TL;DR

This paper extends classical Schubert calculus results to the compactified moduli space of points on P^1, constructing families of intersections with osculating flags and analyzing their topology, especially over real points.

Contribution

It constructs a flat Cohen-Macaulay family over ar{M}_{0,r} for Schubert problems with osculating flags and studies the topology of real solutions, linking to Young tableaux.

Findings

Constructed a flat Cohen-Macaulay family over ar{M}_{0,r}

Extended Schubert intersection results to the compactification

Described the topology of real solutions as a CW complex

Abstract

Given a point z in P^1, 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(z_1), F(z_2), ..., F(z_r) has been studied both classically and recently, especially when the points z_i are real. Since the rational normal curve has an action of PGL_2, it is natural to consider the points (z_1, ..., z_r) as living in the moduli space of r distinct point in P^1 -- the famous M_{0,r}. One can then ask to extend the results on Schubert intersections to the compactification \bar{M}_{0,r}. The first part of this paper achieves this goal. We construct a flat, Cohen-Macaulay family over \bar{M}_{0,r}, whose fibers over M_{0,r} are isomorphic to G(d,n) and, given partitions lambda_1, ..., lambda_r, we construct a flat Cohen-Macualay family over \bar{M}_{0,r} whose fiber over (z_1, ..., z_r) in M_{0,r} is the intersection of the Schubert varieties indexed by lambda_i with respect to the osculating flags F(z_i). In the second part of the paper, we investigate the topology of the real points of our family, in the case that sum |lambda_i| = dim G(d,n). We show that our family is a finite covering space of \bar{M}_{0,r}, and give an explicit CW decomposition of this cover whose faces are indexed by objects from the theory of Young tableaux.