Explicit Noether-Lefschetz for arbitrary threefolds.
The traditional Noether-Lefschetz theorem states that in degree at
least 4 a generic surface in the projective space P3 has no curves
other than complete intersections with other surfaces in P3. Generic
in this sense means outside of a countable union of sub-varieties. The
explicit Noether-Lefschetz theorem is a refinement due to Voisin and
Green independently stating that for surfaces of degree d each of
these "bad" subvarieties is of codimension at least d-3. We will show
how to extend this work to threefolds other than P3.