## Background

I am interested in the projective classification of reduced curves of degree four in $\mathbb{P}^3(\mathbb{R})$ (and more generally of degree $n+1$ in $\mathbb{P}^n(\mathbb{R})$). More precisely, I am looking at the case where the curve is a union of four distinct lines. I need this classification because I want to make sure that I consider all possible cases in a problem in interpolation theory.

For instance, there are two types of configurations of three lines in $\mathbb{P}^2$. Either three lines meet in a single point, or three lines meet in three distinct points. More generally, according to this integer sequence, there are 3 configurations of four lines in $\mathbb{P}^2$, 5 configurations of five lines in $\mathbb{P}^2$, and 18 configurations of 6 lines in $\mathbb{P}^2$. These configurations are shown in this figure (except for the configurations in which all lines are concurrent).

I believe there are six configurations of three lines in $\mathbb{P}^3$: Two configurations for which the three lines lie in a plane, three configurations for which precisely two of the three lines lie in a plane, and one configuration where none of the lines intersect.

My (related) questions are now as follows:

- How many configurations are there of four lines in $\mathbb{P}^3$ (and more generally of $n+1$ lines in $\mathbb{P}^n$)?
- Is there a convenient way to enumerate these?

up to projectivities? Charles Siegel's answer shows that this is not a combinatorial problem. Maybe you want instead to look at configuration of lines up to homotopies that preserve crossing (along the homotopy, lines that initially cross should cross all the time and lines that does not must not cross at any time). – Benoît Kloeckner Aug 13 '10 at 13:00