Reference request: Projective space $\mathbb{RP}^3$

Question

I have to write seminar paper in Projective geometry and the name of the seminar is (I would try to translate):

Representation of the lines in the projective space $\mathbb{RP}^3-$Kernel, Lineal, Pliker's matrix, Pliker's coordinates - subsection.

I tried to search something about Pliker's matix, etc, but I failed.

Any help is appreciated.

Answer 1

"Plücker coordinates", and "Grassmannian" are the right key words for what you are looking for. The object you want is the Grassmannian $G(2,4)$ (or $G(1,\mathbb P^3)$, depending on notations). You can realize this variety, the smallest Grassmannian which is not a projective space, as a quadric hypersurface in $\mathbb P^5$, via the "Plücker embedding".