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 [1] 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 [2] 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: