A geometric application of Nori's connectivity theorem

Abstract

We study (rational) sweeping out of general hypersurfaces by varieties having small moduli spaces. As a consequence, we show that general K -trivial hypersurfaces are not rationally swept out by abelian varieties of dimension at least two. As a corollary, we show that Clemens' conjecture on the finiteness of rational curves of given degree in a general quintic threefold, and Lang's conjecture saying that such varieties should be rationally swept-out by abelian varieties, contradict. Mathematics Subject Classification (2000): 14C05 (primary); 14D07 (secondary).