On threefolds covered by lines

A classification theorem is given of projective threefolds that are covered by the lines of a two-dimensional family, but not by a higher dimensional family. Precisely, if X is such a threefold, let S denote the Fano scheme of lines on X and m the number of lines contained in X and passing through a general point of X. Assume that S is generically reduced. Then m < 6. Moreover, X is birationally a scroll over a surface (m = 1), or X is a quadric bundle, or X belongs to a finite list of threefolds of degree at most 6. The smooth varieties of the third type are precisely the Fano threefolds with - K_X = 2H_X.