Why is $\mathbb{P}(T\mathbb{P}^2)$ isomorphic to the space of complete flags $GL_3/B$? In one the the answers to this thread " Can one embedd the projectivezed tangent space of CP^2 in a projective space?  " it was mentioned that " $\mathbb{P}(T\mathbb{P}^2)$ isomorphic to the variety of complete flags in the vector space $\mathbb{C^3}$ ". 
I'm having a hard time understanding why this is true, and can't seem to find any references. 
 A: I will take $P^2$ to mean the space of lines in $C^3$. The tangent space at a particular point in $P^2$ (say represented by a line $L$) is a linear map from $L$ to $C^3/L$. Let $\mathcal{L}$ be the line bundle whose total space is $\{(L,x) \in P^2 \times C^3 \mid x \in L\}$ (this is $O(-1)$ but it doesn't matter). Then from the first fact, the tangent bundle is $Hom(\mathcal{L}, C^3/\mathcal{L}) \cong \mathcal{L}^* \otimes (C^3/\mathcal{L})$ where I use $C^3$ to be the trivial bundle $C^3 \times P^2$.
The projectivization of a vector bundle ignores tensoring with line bundles, so $P(TP^2)$ is the same as the space of projectivization of $C^3/\mathcal{L})$. So its points correspond to a choice of line $L$ and a choice of line in $C^3/L$. The latter is equivalent to a 2-dimensional subspace in $C^3$ containing $L$.
In general, $P(TP^n)$ is just the space of partial flags of type $(1,2)$ in $C^{n+1}$.
EDIT. A derivation of the above fact for tangent spaces of projective spaces can be found here: http://concretenonsense.wordpress.com/2009/08/17/tangent-bundle-of-the-grassmannian/
A: Let $\pi:\mathbb{P}(T_{\mathbb{P}^2})\rightarrow\mathbb{P}^2$ be the projectivized tangent bundle. The point $x:=(p,[L])\in\mathbb{P}(T_{\mathbb{P}^2})$ corresponds to the point $p = \pi((p,[L]))\in\mathbb{P}^2$ and to the class of the line $L\subset\mathbb{P}^2$ passing through $p$. Now, the point $p$ is a line through the origin $V_p\subset\mathbb{C}^3$, the line $L_p$ corresponds to a plane $\Pi_p\subset\mathbb{C}^3$. Clearly, we have $V_p\subset\Pi_p\subset\mathbb{C}^3$. Now, the morphism
$$\phi:\mathbb{P}(T_{\mathbb{P}^2})\rightarrow F(1,2,3),\: x:=(p,[L])\mapsto (V_p,\Pi_p),$$
is an isomorphism because $\phi$ is injective, and $\mathbb{P}(T_{\mathbb{P}^2})$, $F(1,2,3)$ are both smooth.
