Help about "Varieties with small Dual Varieties" by L.Ein

Question

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!

Answer 1

It seems to me that these various (somehow independent questions) could be asked to your master thesis advisor. It's basically what such an advisor is made for, I guess. A few hints:

1- This is an obvious consequence of the Euler exact sequence which identifies $T_{\mathbb{P}(V)}(-1)|_{\ell}$ with $V/\ell$. See page 40-41 (and before) of the book of Okonek, Scneider, Spindler (vector bundles on complex projective space), where this is explained in some details.

2- $\Omega^2(2) = \bigwedge^2 \Omega(1)$, so the first inclusion is the Plucker embedding in family (but you're missing $\mathbb{P}$ in front of $\Omega_{T}^2(2)$). The answer to your first question gives $I = Gr(2, \Omega^1_T)$ (the Grassmannian is the Grassmannian of quotients),so you are asking twice the same question. You should study the fibers of $E \longrightarrow D$.

3- Apply $Sym^2$ to the dual of the twisted Euler exact sequence. Long exact sequence in cohomology, Bott vanishing Theorem on projective space.

4- The normal bundle of $\mathbb{P}^9$ in $\mathbb{P}^{10}$ is $\mathcal{O}_{\mathbb{P}^9}(1)$ and use the adjunction formula.

5- It's a matter of what you call the "tautological bunde", but fiberwise this is indeed the tauatological bundle.