generic with respect to each other (for example, they could all be equal). This would be most unfortunate; however luckily it does not happen.

Let $\rho^i$ denote the Schubert position of $S \subset V$ with respect to the flag $F^i_V$. Thus $S \in \bigcap_{i=1}^s \Omega^{F^i_V}_{\rho^i}$. We must show that $S$ is in fact a generic point of this intersection of Schubert varieties. This is sufficient, since the induced flags at a generic point of an intersection of Schubert varieties are generic (c.f. remark \[rmk:genericinducedflags\]).

The idea is to fix generic flags $F^i_V$, while the flags $F^i_Q$ vary. We show that there cannot exist a subvariety $T \subset \bigcap_{i=1}^s \Omega^{F^i_V}_{\rho^i}$ such that $\ker \phi \in T$, for a generic choice of $\phi \in \bigcap_{i=1}^s X_{\lambda^i}$, and generic induced flags $F^i_Q$. This is a consequence of a generalisation of the Kleiman moving lemma, due to Belkale.

\[lem:moving\] Let $H$ be an algebraic group acting on a variety $X$. Suppose $\pi: X \to Y$ is an $H$-invariant fibration, such that $H$ acts transitively on the fibres. Let $Z_i \subset X$, and $Y_i=\pi(Z_i) \subset Y$ $i \in \{1, \ldots, s\}$, be subvarieties, such that $\pi|_{Z_i}: Z_i \to Y_i$ is a fibration. Put $Y_0 = 