Enumeration of surfaces containing a curve of low degree

Noether–Lefschetz theory tells us that a very general surface of degree at least 4 in P3 has Picard group Z, i.e.,it contains only curves which are complete intersections with other surfaces. Let W be some irreducible subvariety of the Hilbert scheme of curves in P3. For all sufficiently large d, the surfaces of degree d containing a member of W form a subvariety NL(W,d) of |ØP3(d)|. We give formulas for the degree of NL(W,d), polynomial in d, when the general member C∈W is of one of the following types: a union of up to three general lines, a conic, or a twisted cubic curve.