= X_{{\lambda^i}'}$ are linear subspaces of ${\mathrm{Hom}}(V/S,Q)$ their intersection is necessarily equidimensional. Thus it suffices to show that the intersection is transverse on an open subset which contains a point of intersection.

Consider the space ${\mathrm{Hom}}_0(V/S,Q) = \{\psi \in {\mathrm{Hom}}(V/S,Q)\ |\ \ker \psi = {\{0\}}\}.$ This is a homogeneous space under the action of $GL(V/S) \times GL(Q)$, and it is a Zariski open subset of ${\mathrm{Hom}}(V/S,Q)$. By corollary \[cor:genericdescent\] the $X_{{\lambda^i}'}$ are generic translates of $\hat X_{{\lambda^i}'}$ by elements of $GL(V/S) \times GL(Q)$. So by Kleiman-Bertini, the intersection $$\bigcap_{i=1}^s \left({\mathrm{Hom}}_0(V/S,Q) \cap X_{\lambda^i}\right)$$ is a transverse intersection. Moreover, it contains the point $\tilde \phi$ which shows that the intersection $\bigcap_{i=1}^s X_{{\lambda^i}'}$ is transverse.

Two-step flag manifolds {#sec:twostep}
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### The tangent space to a Schubert variety in a two-step flag manifold

Let $0<d<r<n$. We consider the two-step flag manifold $Fl(d,r,{\mathbb{C}}^n)$. For two-step flag manifolds, we use the notation $\hat{Y}_{\sigma}$ for $\hat{Z}_\sigma$, and $Y_\sigma$ for the generic translate $Z_\sigma$. Here $\sigma$ is a $012$-string, with $d$ [[‘$2$’]{}s ]{}, $r-d$ [[‘$1$’]{}s ]{}, and $n-r$ [[‘$0$’]{}s ]{}. Let $\eta_1 < \eta_2 < \cdots <\eta_{n-r}$ denote the positions of the [[‘$0$’]{}s ]{}(i.e $\sigma_{\eta_m}=0$, for $m \leq n-r$). Let $\eta_{n-r+1} < \cdots < \eta_{n-d}$ denote the 