A General NoetherLefschetz Theorem and applications
Abstract
In this paper we generalize the classical NoetherLefschetz 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 NoetherLefschetz locus of the linear system $H^0(X,L^n)$ is a countable union of proper closed subvarieties of $¶(H^0(X,L^n)^*)$ 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 NoetherLefschetz 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 $S=¶(H^0(X,L^n)^*)$, 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 NoetherLefschetz theorem on codimension one cycles on a smooth projective threefolds to arbitrary codimension
 Publication:

arXiv eprints
 Pub Date:
 May 1993
 arXiv:
 arXiv:alggeom/9305001
 Bibcode:
 1993alg.geom..5001J
 Keywords:

 Mathematics  Algebraic Geometry
 EPrint:
 30 pages, in LaTeX. replaced to correct earlier email corruption