A General Noether-Lefschetz Theorem and applications

Abstract

In this paper we generalize the classical Noether-Lefschetz Theorem to arbitrary smooth projective threefolds. Let $X$ be a smooth projective threefold over complex numbers, $L$ a very ample line bundle on $X$. Then we prove that there is a positive integer $n_0(X,L)$ such that for $n \geq n_0(X,L)$, the Noether-Lefschetz locus of the linear system $H^0(X,L^n)$ is a countable union of proper closed subvarieties of $\P(H^0(X,L^n)^*)$ of codimension at least two. In particular, the {\em general singular member} of the linear system $H^0(X,L^n)$ is not contained in the Noether-Lefschetz locus. As an application of our main theorem we prove the following result: Let $X$ be a smooth projective threefold, $L$ a very ample line bundle. Assume that $n$ is very large. Let $S=\P(H^0(X,L^n)^*)$, let $K$ denote the function field of $S$. Let ${\cal Y}_K$ be the generic hypersurface corresponding to the sections of $H^0(X,L^n)$. Then we show that the natural map on codimension two cycles $$ CH^2(X_{\C}) \to CH^({\cal Y}_K) $$ is injective. This is a weaker version of a conjecture of M. V. Nori, which generalises the Noether-Lefschetz theorem on codimension one cycles on a smooth projective threefolds to arbitrary codimension