I'm studying the paper "Varieties with small Dual Varieties" by L.Ein and in the construction he gives about the $10-$dimensional spinor variety $S_4 \subset \mathbb{P}^{15}$ I'm finding some hard stuff that I'm not able to figure it out. Let me briefly recall the notations. Set $T=\mathbb{P}(W)=\mathbb{P}^4$ and $\mathbb{G}(2,4) \subset D=\mathbb{P}(\bigwedge^3 W )=\mathbb{P}^9$ the Grassmannian of $2-$planes in $\mathbb{P}^4$. Let us consider the incidence variety $$I=\{(p,[\Pi])|p \in T, [\Pi] \in \mathbb{G}(2,4) \: \text{with} \: p \in \Pi\}$$ with the two canonical projections into $T$ and $\mathbb{G}(2,4)$.

**Questions:**

- Why the fiber $I_p \subset I$ over a point $p \in T$ is isomorphic to the set of $2$-dimensional quotient spaces of $\Omega_T^1(1)_p$, where $\Omega_T^1(1)_p$ is the fiber of the vector bundle $\Omega_T^1(1)$ at $p \in T$?
- Why $Gr(2,\Omega_T^1(1)) \subset \Omega_T^2(2)$ and why the inclusion $I \subset \mathbb{P}(\Omega_T^2(2))$ is given by the Plucker embedding? And why in particular $E=\mathbb{P}(\Omega_T^2(2))$ is the blowup of $D$ along $\mathbb{G}(2,4)$, with $I \subset E$ its exceptional divisor?
- Why $H^0(Sym^2(\Omega_T^2(2))) \cong \text{Hom}(\Omega_T^4,Sym^2(\Omega_T^2(2)))$?
- Let us embed $D \subset \mathbb{P}^{10}$ as an hyperplane and consider $\pi:Y \rightarrow \mathbb{P}^{10}$ the blowup along $\mathbb{G}(2,4) \subset D$, with $F \subset Y$ its exceptional divisor. Now clearly $E$ lives naturally inside $Y$ simply because the retriction of $\pi$ to $D$ is just the blowup of $D$ along $\mathbb{G}(2,4)$. My question is the following: why $$\mathcal{L}=\pi^{*}\mathcal{O}_{\mathbb{P}^{10}}(2) \otimes \mathcal{O}(-F)=\pi^{*}\mathcal{O}_{\mathbb{P}^{10}}(1) \otimes \mathcal{O}(-E)$$ and $\mathcal{L}_{|E}=h^*\mathcal{O}_{T}(1)$, where $h:\mathbb{P}(\Omega_T^2(2)) \longrightarrow T$ is the projective bundle structure?
- At a certain point Ein says that $\mathcal{O}_{E}(-E)$ is the tautological line bundle of $\mathbb{P}(\bigwedge^2 \Omega^1_T \otimes \mathcal{O}_T(2))$, and I'm pretty confused about why this is true.

I'm quite familiar with the constructions mentioned in the paper but I find hard to justify many results that Ein gives without further explaining them.

Thank in advance for the help!