Abstract
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, ifX is such a threefold, let Σ denote the Fano scheme of lines onX and μ the number of lines contained inX and passing through a general point ofX. Assume that Σ is generically reduced. Then μ ≤ 6. Moreover,X is birationally a scroll over a surface (μ = 1), orX is a quadric bundle, orX 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 .
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