The Generalized Secant Conjecture for Schubert problems on Grassmannians

Let $G(k,n)$ be a Grassmannian, let $\gamma$ be a rational normal curve, and let $\lambda^1,\dotsc,\lambda^m$ be a Schubert problem on $G(k,n)$. A generalized secant subspace to $\gamma$ is spanned by osculating subspaces of $\gamma$; a flag is generalized secant if each of its subspaces is generalized secant. Assume that $F_\bullet^1,\dotsc,F_\bullet^m$ are generalized secant flags whose spanning osculating subspaces lie along pairwise disjoint intervals of $\gamma$. Generalized Secant Conjecture. The intersection

$$X_{\lambda^1}F_\bullet^1\cap X_{\lambda^2}F_\bullet^2\cap\dotsb\cap X_{\lambda^m}F_\bullet^m$$

is transverse and consists of $d(\lambda^1,\dotsc,\lambda^m)$ real points. This contains the Secant Conjecture when the flags are secant and the Shapiro theorem when they are osculating. Its general validity remains open, although the paper reports computational tests of cases with mixtures of osculating and secant flags.