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Published online by Cambridge University Press: 01 March 2000
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: