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Schubert problems with respect to osculating flags of stable rational curves
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
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