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

Questions:

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!