I am following section 4 of chapter 1 of Griffiths-Harris, "Principles of Algebraic Geometry", and I want to prove that every algebraic curve embeds in $\mathbb{P}^3(\mathbb{C})$. If you know a simple proof of this fact, I would also appreciate a reference to that. Anyway, the book defines the $\textit{chordal variety}$ (denoted with $C(V)$) of an algebraic variety $V \subset \mathbb{P}^n$ as
$\textit{the union of all the lines that meet V twice and the lines tangent to V}$.
Then, it is stated that $C(V)$ is the image of the projection on the third factor of the closure of the incidence correspondence, $I\subset \mathbb{P}^n \times \mathbb{P}^n \times \mathbb{P}^n$, defined as $$I=\{(p,q,r)| p \neq q \in V, p\wedge q \wedge r=0\}.$$ I am not sure I've understood the definition of incidence correspondence (I mean, those conditions look pretty obscure), but for sure I've not understood why $C(V)$ is the image of the closure of $I$ under the projection on the third factor. Any help (also references) will be appreciated.