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Vakil’s FOAG gives the definition of the dual peojective space via introducing new indeterminates: (sorry that I have to quote it as a screenshot) enter image description here

And the answer in Scheme theoretic dual of $\mathbb P^n_k$ gives a coordinate free definition, as $(\Bbb{P}V)^\vee = \Bbb{P}(V^\vee )$. How is this coordinate free definition connect to the relation $a_0 x_0 + a_1 x_1 + \cdots a_n x_n = 0$? Given $a_i$ the dual base for $x_i$, shouldn’t it be $a_i x_j =\delta_{ij}$?

Thank you in advance.

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    $\begingroup$ The incidence variety is $I\subset\mathbb P(V)\times\mathbb P(V^\vee)$ consisting of $(v,v^\vee)\in \mathbb P(V)\times\mathbb P(V^\vee)$ such that $v^\vee(v)=0$, which is well-defined since the condition is invariant under constant multiplication. Now writing the equation in terms of bases recovers Vakil's description. $\endgroup$
    – Kenta S
    Commented May 27, 2023 at 22:33
  • $\begingroup$ @KentaS that looks like an answer to me - please consider recording it as such below. $\endgroup$
    – KReiser
    Commented May 28, 2023 at 20:55

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The incidence variety is $I\subset\mathbb P(V)\times\mathbb P(V^\vee)$ consisting of $(v,v^\vee)\in\mathbb P(V)\times\mathbb P(V^\vee)$ such that $v^\vee(v)=0$, which is well-defined since the condition is invariant under constant multiplication. Now writing the equation in terms of bases recovers Vakil's description.


In terms of the Proj construction, $\mathbb P(V)=\mathrm{Proj}(\mathrm{Sym}^\bullet V^\vee)$ and $\mathbb P(V^\vee)=\mathrm{Proj}(\mathrm{Sym}^\bullet V)$, so $$\mathbb P(V)\times \mathbb P(V^\vee)=\mathrm{Proj}(\bigoplus_{n\ge0}\mathrm{Sym}^nV^\vee\otimes \mathrm{Sym}^n V).$$ Now the incidence variety is defined by the ideal generated by the element $\mathrm{id}_{V}\in\mathrm{hom}(V,V)\cong V\otimes V^\vee$, which gives an element of degree $1$ in $\bigoplus_{n\ge0}\mathrm{Sym}^nV^\vee\otimes \mathrm{Sym}^n V$.

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  • $\begingroup$ Thank you for the answer. How to get the equation $\mathbb P(V)\times \mathbb P(V^\vee)=\mathrm{Proj}(\mathrm{Sym}^\bullet V^\vee\otimes \mathrm{Sym}^\bullet V)$? I sawed that product of two Proj does not concide with Proj of the two tensor proudct here math.stackexchange.com/questions/1616432/… $\endgroup$
    – onRiv
    Commented May 29, 2023 at 0:26
  • $\begingroup$ Sorry you are right, I will fix this. $\endgroup$
    – Kenta S
    Commented May 29, 2023 at 0:29
  • $\begingroup$ I think I understand it now. It’s a nice and elegant coordinate free description for the incident variety. $\endgroup$
    – onRiv
    Commented May 29, 2023 at 0:48
  • $\begingroup$ And would you mind taking a look at the lined question I posted? $\endgroup$
    – onRiv
    Commented May 29, 2023 at 1:04

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