One-dimensional Schubert problems with respect to osculating flags

Abstract: We consider Schubert problems with respect to flags osculating the rational normal curve. These problems are of special interest when the osculation points are all real -- in this case, for zero-dimensional Schubert problems, the solutions are "as real as possible". Recent work by Speyer has extended the theory to the moduli space $\overline{M_{0,r}}$, allowing the points to collide. These give rise to smooth covers of $\overline{M_{0,r}}(\mathbb{R})$, with structure and monodromy described by Young tableaux and jeu de taquin. In this paper, we give analogous results on one-dimensional Schubert problems over $\overline{M_{0,r}}$. Their (real) geometry turns out to be described by orbits of Sch\"{u}tzenberger promotion and a related operation involving tableau evacuation. Over $M_{0,r}$, our results show that the real points of the solution curves are smooth. We also find a new identity involving `first-order' K-theoretic Littlewood-Richardson coefficients, for which there does not appear to be a known combinatorial proof.