# 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!

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.

• thank you very much for the response! I don't get the point 4), namely sections of $\pi^*\mathcal{O}_{\mathbb{P}^{10}}(1) \otimes \mathcal{O}(-E)$ correspond to hyperplanes of $\mathbb{P}^{10}$ containing $D$, which is a $\mathbb{P}^9$. This has to have only one section, i.e. $D$, I don't quite get how $\pi^*\mathcal{O}_{\mathbb{P}^{10}}(1) \otimes \mathcal{O}(-E)=\pi^*\mathcal{O}_{\mathbb{P}^{10}}(2) \otimes \mathcal{O}(-F)$, since the latter has plenty of sections...
– gigi
Jun 9, 2022 at 11:55