Universal family of grassmannian as projective bundle over $\mathbb P^n$ Let $p:X=\mathbb P(\mathcal E_2)\rightarrow Gr(2,n+1)$ be the universal family of the lines in $\mathbb P^n$. If we denote $e:X\rightarrow \mathbb P^n$ the natural projection, we have $\mathcal O_{\mathcal E_2}(1)=e^*\mathcal O_{\mathbb P^n}(1)$.
It seems that $X$ is isomorphic to $\mathbb P(T_{\mathbb P^n})$, $e$ being the projection, and that $\mathcal O_{T_{\mathbb P^n}}(1)=p^*\mathcal O_{Gr(2,n+1)}(1)$ where $\mathcal O_{Gr(2,n+1)}(1)$ is the polarization given by the Plücker embedding.
But it seems not to be true since when I compute the first Chern class of $X$ using the first description I get $c_1(T_X)=(n+2)p^*c_1(\mathcal O_{Gr(2,n+1)}(1)) + 2e^*c_1(\mathcal O_{\mathbb P^n}(1))$ whereas using the second one I get $c_1(T_X)=np^*c_1(\mathcal O_{Gr(2,n+1)}(1))+ 2(n+1)c_1(\mathcal O_{\mathbb P^n}(1))$. So, where is my mistake?
 A: The variety $X$ is the partial flag variety $Fl(1,2;n+1)$. With respect to the projection to $P^n$ it is most naturally written as the projectivization of the bundle $T(-2)$. The reason is the following.
Denote by $V$ the vector space of dimension $n+1$, so that $P^n = P(V)$. Then on $P(V)$ one considers the Euler sequence
$$
0 \to O(-1) \to V \otimes O \to T(-1) \to 0.
$$
Its exterior square then is the sequence
$$
0 \to T(-2) \to \Lambda^2V \otimes O \to \Lambda^2T(-2) \to 0.
$$
Thus, $T(-2)$ canonically embeds into the trivial bundle with fiber $\Lambda^2V$, and this embedding induces a morphism $P(T(-2)) \to P(\Lambda^2V)$ that factors through $Gr(2,V) \subset P(\Lambda^2V)$.
EDIT. Let me describe the construction of the first map in the second sequence. First, note that we have a natural map 
$$
V \otimes O(-1) \to V \otimes V \otimes O \to \Lambda^2V \otimes O
$$
(the first map is the identity of $V$ tensored with the canonical embedding of $O(-1)$; the second is the alteration).
It is easy to see that the composition of this map with the embedding $O(-2) \to V \otimes O(-1)$ (the $O(-1)$-twist of the canonical embedding) is zero (because of the alteration). Therefore the map $V \otimes O(-1) \to \Lambda^2V \otimes O$ factors through $Coker(O(-2) \to V \otimes O(-1)) \cong T(-2).$ 
