ag.algebraic geometry – Help about “Varieties with small Dual Varieties” by L.Ein

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 $mathbb{G}(2,4)$.


  1. 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 $p in T$?
  2. 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?
  3. Why $H^0(Sym^2(Omega_T^2(2))) cong text{Hom}(Omega_T^4,Sym^2(Omega_T^2(2)))$?
  4. 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?
  5. 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!