Let $\mathcal{V}= 0 \subset V\_1 \subset \cdots \subset V\_{n-1}\subset V\_n=V$, $\mathcal{W}=0 \subset W\_1 \subset \cdots \subset W\_{n-1} \subset W\_n=W$ be two flags. We say that $\mathcal{V}$ and $\mathcal{W}$ are transverse if $V\_i \cap W\_{n-i}=0$ for all $i$. Now consider the following theorem:


**Theorem** (Kleiman's Theorem in characteristic 0)


Suppose that an irreducible algebraic group $G$ acts transitively on a variety $X$ over an algebraically closed field of characteristic $0$, and that $A \subset X$ is a subvariety.


1. Is $B \subset X$ is another subvariety, then there is an open dense set of $g$ such that $gA$ is generically transverse to $B$.
2. If $G$ is affine, then $[gA]=[A]$ in $A(X)$ (the Chow Ring of $X$) for any $g \in G$.


Now consider two Schubert cycles $\Sigma\_{a}(\mathcal{V})$, $\Sigma\_{b}(\mathcal{W})$, with respect to two transverse flags $\mathcal{V},\mathcal{W}$. Can i deduce from Kleiman's theorem that the two cycles intersect generically transversely whenever $\mathcal{V}$ and $\mathcal{W}$ are transverse? If yes, why?


This is what Harris and Eisenbud claim in their book "3264 and all that" (see pag. 108, the last two lines).



Schubert cycles that intersect generically transversely
By generic transversality, there is an open dense set $U$ in $\mathrm{GL}\_n$ such that $\Sigma\_a(V)$ is transversal to $\Sigma\_b(gV)=g\Sigma\_b(V)$. There is another open dense set $V$ in $\mathrm{GL}\_n$ such that $gV$ is transversal to $V$. It follows that there is a $g$ such that $\Sigma\_a(V)$ is transverse to $\Sigma\_b(W)$ with $W=gV$ *also* transverse to $V$.


Any pair of transverse flags $V$ and $W$ determines a basis (upto scalar multiple) as $V\_i\cap W\_{n-i+1}=k\cdot e\_i$. Conversely, this basis determines $V$ (respectively $W$) as the increasing (respectively decreasing) span of the $e\_i$'s. Thus, if $(V,W)$ is one pair of transverse flags, it can be taken to another pair $(V',W')$ by a change of basis.


Obviously, such a change of basis does not affect the transversality of the intersection of $\Sigma\_a(V)$ and $\Sigma\_b(W)$ as it (the change of basis) applies to both factors.


