# Two definitions of scheme theoretic dual projective space

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)

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 $$a_i x_j =\delta_{ij}$$?

• 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. Commented May 27, 2023 at 22:33
• @KentaS that looks like an answer to me - please consider recording it as such below. Commented May 28, 2023 at 20:55

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 $$\bigoplus_{n\ge0}\mathrm{Sym}^nV^\vee\otimes \mathrm{Sym}^n V$$.
• 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/… Commented May 29, 2023 at 0:26