Let k be an algebraically closed field of arbitrary characteristic. Lines in ℙ3 are
parametrized by the Grassmannian G(2, 4), which is isomorphic to a smooth quadric
in ℙ5. We can consider the configuration space Xn =
G(2, 4)n / PGL4(k) parametrizing
ordered n-tuples of lines in ℙ3 up to projective equivalence.
dim PGL4(k) = 15 and for n [ges ] 5, the stabilizer
of a general n-tuple of lines is trivial, so for n [ges ] 5,
Xn has the expected dimension 4n − 15.
The question of rationality of Xn was posed by Dolgachev.
The space Xn is clearly unirational, since there is a dominant
rational map to it from the rational variety G(2, 4)n. The
following results are known in characteristic 0: it is a special case of a
theorem by Dolgachev and Boden  for configuration spaces in greater generality
that if Xn is rational for some n [ges ] 5 then so is
XN for any N [ges ] n. They also proved
that the configuration space of lines in ℙm is rational
if m is odd and recently Zaitsev  proved this for all m.
Our proof uses different methods and it also has the advantage that it is valid in
any characteristic. The main result is the following: