Up to projectivities, which configurations of four lines in $\mathbb{P}^3$ can one distinguish? 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?
  

 A: This is one of my favourite projective geometry examples; it is a case of
classification involving moduli and if done right most of the arguments can be
done with a combination of geometry and linear algebra with no explicit
calculation.
I assume that we are talking about an ordered quadruple of lines in $\mathbb
P^3(k)$ (I do it over any field). The trick is to not think of $\mathbb P^3$ as
the set of lines in $k^4$ but rather of lines in some $4$-dimensional vector
space V and then use the date to get closer to an adapted coordinatisation.
Assume first that no two of the lines are skew. Then a simple geometric argument
shows that either all lines lie in a plane or pass through a common point. Both
of those cases are reduced to the problem of four points in $\mathbb P^2$ which
I skip. We can then assume that the first two lines are skew and think of $V$ as
$V_1\bigoplus V_2$, where the projectivisations of $V_1$ and $V_2$ are the two
first lines. Assume then that the third line is skew with the first and
second line. This means that it is the projectivisation of the graph $V_3\subset
V_1\bigoplus V_2$ of an isomorphism. Hence, we may assume that $V_1=V_2$ and
$V_3$ is the diagonal in $V_1\bigoplus V_1$. If we also assume that the fourth
line is skew with the first two, then it is also the graph $V_4\subset
V_1\bigoplus V_1$ of an automorphism $V_1\rightarrow V_1$. Hence, four lines,
the first two of which are skew and the last two are skew with the first two up
to projective transformations correspond to isomorphism classes of pairs
$(V_1,\varphi)$ where $V_1$ is a two-dimensional vector space and $\varphi$ is
an automorphism of it distinct from the identity. This is the same thing as
conjugacy classes of $\mathrm{GL}_2(k)$ distinct from the identity element. The
condition that the last two lines be skew is exactly that $\varphi$ does not
have $1$ as an eigenvalue. Of course the characteristic polynomial distinguish
between conjugacy classes so there are continuous families of configuration
(i.e., it has non-trivial moduli).
The remaining case of two skew lines and two lines which are not both skew with
respect to both of the first lines is easy but a little bit tedious; given a
pair of non skew lines one looks at the plane spanned by them and the position
of the other lines with respect to it.
Addendum: I did not mean to suggest that it is the simplest problem with proper moduli. Of course the classification of four points in $\mathbb P^1$ has the cross ratio is its moduli. (The classification is proven almost word for word in the same way as the above case.)
A: 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 $n^2-2n-2$.
