On some line congruences in P^4
Consider a line congruence in P4(C) or, equivalently, a smooth threefold V in the Grassmannian G(1,4); We say that the congruence has type (m,n) if V is numerically equivalent to mΩ(0,4)+nΩ(1,3).
We prove that there are no general, non-degenerate line congruences of type (m,1), for any m, and (m,2) for m≤5.
Further, we give an explicit example of a general line congruence in P4(C), which is a generalization of the classical Reye congruence in P3(C), and we show that its type is (15,10).
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