In one the the answers to this thread " [Can one embedd the projectivezed tangent space of CP^2 in a projective space?](https://mathoverflow.net/questions/119850/)  " 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. 



Why is $\mathbb{P}(T\mathbb{P}^2)$ isomorphic to the space of complete flags $GL_3/B$?
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.


