A Noether-lefschetz Theorem and Applications

In this paper we generalize the classical Noether-Lefschetz Theorem (see [7], [5]) to arbitrary smooth projective threefolds. More specifically, we prove that given any smooth projective threefold X over complex numbers and a very ample line bundle L on X, there is an integer n0(X,L) such that if n ≥ n0(X,L) then the Noether-Lefschetz locus of the linear system H(X,L) is a countable union of proper closed subvarieties of P(H(X,L)) of codimension at least two. In particular, the general singular member of the linear system H(X,L) is not contained in the NoetherLefschetz locus. This generalizes the results of [5]. In [11], we find a conjecture due to M. V. Nori, which generalizes the Noether-Lefschetz theorem for codimension one cycles on smooth projective threefolds to higher codimension cycles on arbitrary smooth projective varieties. As an application of our main theorem we prove a result which can be thought of as a weaker version of Nori’s conjecture for codimension two cycles on smooth projective threefolds. The idea of the proof is borrowed from an elegant paper of M. Green ([4]) and the work of M. V. Nori (see [11]). In [4] it was shown that the classical Noether-Lefschetz theorem can be reduced to a coherent cohomology vanishing result and the required vanishing was also proved for P. Though