\bigcap_{i=1}^s Y_i$, and $X_0 = \pi^{-1}(Y_0)$. Let $T \subset Y_0$, be a subvariety. Then for generic $h_i \in H$, the intersection $$\pi^{-1}(T) \cap \left ( \bigcap_{i=1}^s h_i \cdot Z_i \right )$$ has the expected dimension as an intersection inside $X_0$. That is, $$\label{eqn:moving}
{\mathop{\mathrm{codim}}}\pi^{-1}(T) \cap \left ( \bigcap_{i=1}^s h_i \cdot Z_i \right )
= {\mathop{\mathrm{codim}}}\pi^{-1}(T) + 
\sum_{i=1}^s {\mathop{\mathrm{codim}}}(Z_i \cap X_0)$$ (here ${\mathop{\mathrm{codim}}}$ means “codimension inside $X_0$”).

In this example, $$X = {\mathrm{Hom}}_d(V,Q) 
= \{\psi \in {\mathrm{Hom}}(V,Q)\ |\ \dim \ker \psi = d\},$$ $$Y = Gr(n-d,V), \qquad \pi(\psi) = \ker(\psi)$$ The group $H$ is $GL(Q)$, which we note preserves the kernel of $\psi \in Z_i$. The action of $GL(Q)$ acts transitively on the induced flags $F^i(Q)$.

The subvariety $Z_i \subset X_{\lambda^i} \subset X$ consists of those elements of $X_{\lambda^i}$ whose kernel is in Schubert position $\rho^i$. Thus a generic translate $h_i \cdot Z_i$ is simply $X_{\lambda^i}$ for a different generic choice of flags $F^i_Q$. Thus the image $\pi(\psi)$ of a generic point in the intersection $\psi \in \bigcap_{i=1}^s h_i \cdot Z_i$, is just the kernel of a generic element of $\bigcap_{i=1}^s X_{\lambda^i}$.

The image of $Z_i$ under $\pi$ is the open Schubert cell $Y_i = \Omega^{F^i_V}_{\rho^i}$. We wish to show that 