is non-zero. Equivalently, a Schubert problem is non-vanishing if and only if the general translates $\bar{\Omega}^{F_1}_{\sigma_1}, \ldots, 
\bar{\Omega}^{F_s}_{\sigma_s}$ of these Schubert varieties have a non-empty generically transverse intersection. This observation allows one to study the vanishing question inside the tangent space to the $G/P$.

In an effort to better understand the geometry behind Horn’s conjecture, we study the tangent spaces of Schubert varieties of two-step flag manifolds, and show how these are related to the problem. The two-step flag manifold $$Fl(d,r,{\mathbb{C}}^n) = \{ (S,V)\ |\ S \subset V \subset {\mathbb{C}}^n,
\ \dim S = d,\ dim V = r\}$$ fibres over the Grassmannian $Gr(r,{\mathbb{C}}^n)$, with fibre $Gr(d,V)$ at the a point $V \in Gr(r,{\mathbb{C}}^n)$. A Schubert problem on $Gr(r,{\mathbb{C}}^n)$ can be “lifted” in a number of different ways to a Schubert problem on $Fl(d,r,{\mathbb{C}}^n)$, in such a way that a non-empty transverse intersection on $Gr(r,{\mathbb{C}}^n)$ lifts to a non-empty transverse intersection on $Fl(d,r,{\mathbb{C}}^n)$. Each way of lifting corresponds to a non-vanishing Schubert problem inside the fibre $Gr(d,V)$. However, it is sometimes possible to see that the intersection on $Fl(d,r,{\mathbb{C}}^n)$ is non-transverse for some trivial reasons. These trivial conditions are seen to be the Horn inequalities.

The difficult part of the 