Let $G=\mathbb{G}(1,\mathbb{P})$ be the Grassmannian variety of lines in $\mathbb{P}^3$. I have to do an affine stratification of $G$. In order to do this we consider the flag $\mathcal{F}$ of the form $p \in L \subset H \subset \mathbb{P}^3$, where $L$ and $H$ are a line and a plane in $\mathbb{P}^3$. Then we define $\Sigma_{a,b}$ the set of lines meeting the $(2-a)$-dimentional plane of $\mathcal{F}$ in a point and the $(3-b)$-dimentional plane of $\mathcal{F}$ in a line. So we have (if $\Lambda \in \mathbb{G}(1,\mathbb{P})$)
$\Sigma_{2,2}=\{L\} \subset \Sigma_{2,1}=\{\Lambda|p \in \Lambda \subset H\} \subset \Sigma_{2,0}=\{\Lambda|p \in \Lambda\} \subset \Sigma_{1,0}=\{\Lambda | \Lambda \cap L \ne \emptyset\} \subset \mathbb{G}(1,\mathbb{P}).$
Now I define $\tilde{\Sigma}_{a,b}$ to be the complement in $\Sigma_{a,b}$ of the union of all other Schibert cycles properly contained in $\Sigma_{a,b}$. So I have to prove that $\tilde{\Sigma}_{a,b}$ is isomorphic to an affine space. \In particular $$\tilde{\Sigma}_{2,0}=\Sigma_{2,0} \setminus \Sigma_{2,1} =\{\Lambda|p \in \Lambda \mbox{ but } p \in \Lambda \subset H \}.$$ How can I build an isomorphism from $\tilde{\Sigma}_{2,0}$ to some affine space?
$\Sigma_{2,0} \cong \Bbb P^2$, and $\Sigma_{2,1} \cong \Bbb P^1\subset\Bbb P^2$, so $\tilde\Sigma_{2,0} \cong \Bbb A^2\subset\Bbb P^2$.