The dimension of the space of lines in projective space.

Question

For example in three dimensional projective space. I thought lines in projective space corresponded to planes in euclidian space. So the space is just the dimension of the Grassmanian $G(2,3)$. Which is $3$. But in a book I'm reading it is said that the dimension must be $4$. What did I do wrong?

Answer 1

You are right that lines in $\mathbb{P}^3$ correspond to planes through the origin in an euclidian space. This is because $\mathbb{P}^3$ is a quotient space of such an euclidian space.

However, remember that $\mathbb{P}^3$ is a quotient of an euclidian space of one dimension higher, i.e. a 4-dimensional euclidian space in this case.

Hence the grassmanian you are looking for is $G(2,4)$ instead of $G(2,3)$.

In general lines in $\mathbb{P}^n$ correspond to $G(2,n+1)$.

Even more general linear spaces of dimension $l$ in $\mathbb{P}^n$ correspond to $G(l+1,n+1)$.