Let $p:P\rightarrow G(2,n+1)$ be the universal $\mathbb P^1$-bundle over the grassmannian. Let us denote $I=P\times_{\mathbb P^n}P$ the incidence correspondence whose generic point is of the form $([l],[l'], x=l\cap l').$ There are three projections that we shall denote $p_1:I\rightarrow G(2,n+1)$, $p_2:I\rightarrow G(2,n+1)$ and $q:I\rightarrow \mathbb P^n$.
There is a map $\varphi:I\dashrightarrow G(3,n+1)$ and we can resolve the indeterminacies by blowing-up the diagonal in $I$ to get a morphism $\tilde \varphi:\tilde I\rightarrow G(3,n+1)$.
According to this answer, we have a morphism $f:\mathbb P(\tilde\varphi^*\mathcal E^{\vee}_3(1))\rightarrow G(2,n+1)$. I would like to compute $c_1(\tilde\varphi^*\mathcal E^{\vee}_3(1))$ and it seems that we have an exact sequence (using the evaluations and the fact that the lines bundles $p_i^*\mathcal O_{G(2,n+1)}(1)$ should give sections of $f$) $$\tilde\varphi^*\mathcal E^{\vee}_3(1)\rightarrow p_1^*\mathcal O_{G(2,n+1)}(1)\oplus p_2^*\mathcal O_{G(2,n+1)}(1)\rightarrow \mathcal L\rightarrow 0$$ where $\mathcal L$ is a sheaf supported on the exceptional divisor of the blow-up $\tilde I\rightarrow I$. I wanted to be sure of this part of the exact sequence and of the kernel of the first morphism to be able to compute $c_1(\tilde\varphi^*\mathcal E^{\vee}_3(1))$.