Background
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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.




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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](http://www.research.att.com/~njas/sequences/A132346), 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](http://www.research.att.com/~njas/sequences/a048872.pdf) (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:



> 
> 1. How many configurations are there of four lines in $\mathbb{P}^3$ (and more generally of $n+1$ lines in $\mathbb{P}^n$)?
> 2. Is there a convenient way to enumerate these?
> 
> 
> 



Up to projectivities, which configurations of four lines in $\mathbb{P}^3$ can one distinguish?
Up to projectivities, there are uncountably many configurations. Let's do the naive dimension count: The Grassmannian of lines in $\mathbb{P}^3$ is four dimensional, so the parameter space for four lines is 16 dimensional. The automorphism group of $\mathbb{P}^3$, the projections, is made up of four by four matrices modulo the diagonal matrices, so is dimension 16-1=15. So we should get a whole curve worth of possible configurations.


This is analagous to how you can use cross ratio to distinguish different configurations of four points in $\mathbb{P}^1$.


EDIT: To make this more general and distinguish it from jvp's comment, if you look at $n+1$ lines in $\mathbb{P}^n$, then you have $(n+1)^2-1=n^2+2n$ automorphisms, and the space of lines is $2(n-1)$ dimensional, so in general you have $2(n-1)(n+1)-(n+1)^2+1$ dimensions worth of configurations, and this simplifies to $n^2-2n-2$.


