there is no subvariety $T \subset \bigcap Y_i$ such that $\ker \psi \in T$ for all choices above. In other words, for any subspace $T$ we can choose $\psi$ and generic flags $F^i_Q$ such that $\psi \in \bigcap_{i=1}^s X_{\lambda^i}$ and $\ker \psi \notin T$. Equivalently, we must show there exists $\psi \in \bigcap h_i \cdot Z_i$, with $\psi \notin \pi^{-1}(T)$. But this is clear from lemma \[lem:moving\], since otherwise the codimensions in equation (\[eqn:moving\]) would not add.

\[cor:genericdescent\] Let $S = \ker \phi$ for a generic $\phi \in \bigcap_{i=1}^s 
X_{\lambda^i}$. Then the $X_{\lambda^i}{\big /}S = X_{{\lambda^i}'}$ are in general position. Moreover there is an element $\tilde \phi \in \bigcap_{i=1}^s X_{{\lambda^i}'}
\subset {\mathrm{Hom}}(V/S,Q)$ such that $\ker \tilde \phi = {\{0\}}$.

The fact that the $X_{\lambda^i}{\big /}S$ are generic simply means that the induced flags $F^i_{V/S}$ and $F^i_{Q}$ are generic, which is what we have just shown. For the second statement, since $\ker \phi = S$, $\phi$ descends to well defined map $\tilde \phi:V/S \to Q$ with $\tilde \phi \in \bigcap_{i=1}^s X_{\lambda^i}{\big /}S$ and $\ker \tilde \phi = {\{0\}}$.

\[cor:transversereduction\] The intersection $\bigcap_{i=1}^s X_{\lambda^i}{\big /}S$ is transverse.

The argument can be seen as special case of [@B]\[lemma 2.18\].

Since the $X_{\lambda^i}{\big /}S 