  One of the themes in algebraic geometry is the study of the relation between
the ``topology'' of a smooth projective variety and a (``general'') hyperplane
section. Recent results of Nori produce cohomological evidence for a conjecture
that a general hypersurface of sufficently large degree should have no
``interesting'' cycles. We compute precise bounds for these results and show by
example that there are indeed interesting cycles for degrees that are not high
enough. In a different direction Esnault, Nori and Srinivas have shown
connectivity for intersections of small multidegree. We show analogous
cycle-theoretic connectivity results.
