# incident variety as the projectivization (relative proj of symmetric sheaf) of some locally free sheaf on the dual projective space

For simplicity let's suppose $$n=2$$ and $$\Bbb{P}^{n}_k = \operatorname{Proj}k[X,Y,Z]$$ and $$\Bbb{P}^{n\vee}_k = \operatorname{Proj}k[A,B,C]$$. With the segre embeding

$$\operatorname{Proj}k[X,Y,Z] \times \operatorname{Proj}k[A,B,C]\to\operatorname{Proj}k[AX,AY,\cdots, CZ]$$

and define "the incident variety" $$\Sigma$$ (or, "the incident correspondence", only find this terminology in Vakil's FOAG,No. 13.4.1) be the closed subscheme in $$\operatorname{Proj}k[AX,AY,\cdots, CZ]$$ vanishing $$AX+BY+CZ$$, i.e.,

$$\Sigma := V(AX+BY+CZ) = \operatorname{Proj}\frac{k[AX,AY,\cdots, CZ]}{AX+BY+CZ}$$

Prove that $$\Sigma \to \operatorname{Proj}k[A,B,C]$$ is the projectivization of some locally free sheaf of rank 2 on $$\operatorname{Proj}k[A,B,C]$$, i.e., there is some locally free sheaf $$\mathscr{E}$$of rank 2 on $$\operatorname{Proj}k[A,B,C]$$, such that $$\Sigma \to \operatorname{Proj}k[A,B,C]$$ is the relative proj of $$\operatorname{Sym}\mathscr{E}$$:

$$\Sigma = \mathcal{Proj}\operatorname{Sym}\mathscr{E} \to \operatorname{Proj}k[A,B,C]$$

I saw this question from David Eisenbud, Joe Harris, The geometry of schemes, Exercise III-25, Page104

The background for it is that I am continuing trying to gain some understanding the scheme theoretic dual projective space $$\Bbb{P}_k^{n\vee}$$, following a question I posted Two definitions of scheme theoretic dual projective space.

The following is what I have tried:

Looking at it in the affine open subscheme $$\operatorname{Spec} k[U,V]$$ where $$U=B/A, V=C/A$$, I got

$$\operatorname{Proj}\frac{k[AX,AY,\cdots, CZ]}{AX+BY+CZ} \to \operatorname{Spec} k[U,V]$$ Then "since $$A \neq 0$$" (this makes no sense scheme theoretically, I just want to show what I have tried), I got (from no reason) $$\operatorname{Proj}\frac{k[U,V][X,Y,Z]}{X+UY+VZ} \to \operatorname{Spec} k[U,V]$$

Then I failed to describe the left hand side of the above map as the symmetric sheaf of some locally free sheaf over $$\operatorname{Spec} k[U,V]$$, making less gluing them into a locally free sheaf on $$\operatorname{Proj}k[A,B,C]$$.

Is the above argument a right direction? If so, how to make it rigorous and conitue the argument? As in the comment of the answer here, the sheaf $$\mathscr{E}$$ seems to be some "equivalent class (maybe quotient out $$\mathscr{O^*}$$?) of the sheaf $$\mathscr{O}(1)$$", in such a way it corresponds to the inutition "A point of the dual projective space is an equivalence class of linear forms on the original vector space, which defines a hyperplane in the usual way." that Zhen Lin said. But I failed to make it concide with the above argument.

Thanks for reading, any help would be appreciated.

• Your $n$s and your variables don't match up - $\operatorname{Proj} k[x,y,z]\cong\Bbb P^2$, not $\Bbb P^3$. Also, one tip for organization: it would be nice to put the problem before your work - it seems that in this post you start working on things first and then say what the problem is, and that's a little harder to follow. May 28, 2023 at 21:12
• @KReiser Thank you. I fixed some typos and reorganized the text. May 29, 2023 at 0:14

By replacing $$X$$ to $$-UY-VZ$$, the scheme $$\operatorname{Proj}\frac{k[U,V][X,Y,Z]}{X+UY+VZ}$$ in fact is just $$\operatorname{Proj}k[U,V][Y,Z]$$. Hence on the affine open subscheme $$\operatorname{Spec} k[U, V]$$, the incident variety $$\Sigma$$ is the projectivization of the rank 2 free sheaf with generators $$Y$$ and $$Z$$.
Changing $$U, V$$ back to $$B/A$$ and $$C/A$$, the transition map from $$\operatorname{Spec} k[B/A, C/A]$$ to $$\operatorname{Spec}k[A/B, C/B]$$, by $$Y = -A/B \cdot X -C/B \cdot Z$$, is
\begin{aligned} \left(\begin{array}{l} Y \\ Z \end{array}\right) & =\left(\begin{array}{c} -\frac{A}{B} X-\frac{C}{B} Z \\ Z \end{array}\right) \\ & =\left(\begin{array}{cc} -\frac{A}{B} & -\frac{C}{B} \\ 0 & 1 \end{array}\right)\left(\begin{array}{l} X \\ Z \end{array}\right) \end{aligned}