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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?

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$\Sigma_{2,0} \cong \Bbb P^2$, and $\Sigma_{2,1} \cong \Bbb P^1\subset\Bbb P^2$, so $\tilde\Sigma_{2,0} \cong \Bbb A^2\subset\Bbb P^2$.

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  • $\begingroup$ Dear @TedShifrin, could you solve the problem using $2 \times 4$ matrices that represent grassmannian? $\endgroup$ – ArthurStuart Dec 10 '13 at 21:41
  • $\begingroup$ Well, you have local representations by $2\times 4$ matrices with certain properties, yes. If $p\leftrightarrow (1,0,0,0)\in\Bbb A^4$ and $H=\text{Span}\big((1,0,0,0),(0,1,0,0),(0,0,1,0)\big)$, then after change of basis we have $$\Lambda\leftrightarrow\begin{bmatrix} 1&0&0&0\\0&*&*&1\end{bmatrix}\,$$ which shows that such $\Lambda$ correspond to $\Bbb A^2$. $\endgroup$ – Ted Shifrin Dec 10 '13 at 22:27
  • $\begingroup$ Dear @TedShifrin, could you give me some reference about examples of affine stratification? Thank you $\endgroup$ – ArthurStuart Dec 10 '13 at 22:31
  • $\begingroup$ Sorry, I actually don't know elementary references. Stratifications and generic conditions on them go back to Whitney. You might try Goresky/MacPherson's Stratified Morse Theory. A wonderful example to work out is the Whitney umbrella, given by $x^2=y^2z$ in $\Bbb A^3$. $\endgroup$ – Ted Shifrin Dec 10 '13 at 22:38

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