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

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
$V_1bigoplus V_2$, 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 $V_3subset V_1bigoplus V_2$ of an isomorphism. Hence, we may assume that $V_1=V_2$ and
$V_3$ is the diagonal in $V_1bigoplus V_1$. If we also assume that the fourth
line is skew with the first two, then it is also the graph $V_4subset V_1bigoplus V_1$ of an automorphism $V_1rightarrow V_1$. 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.)

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