  Let $Y$ be a smooth complex projective variety. We study the cohomology of
smooth families of hypersurfaces $X\to B$ for $B\subset{\bf P}H^0(Y,O(d))$ a
codimension $c$ subvariety. We give an asymptotically optimal bound on $c$ and
$k$ for $d\to\infty$ for the space $H^k(X,\C)$ not to be spanned by the image
of $H^k(Y\times B,\C)$, thus extending the validity of Lefschetz Hyperplane
section Theorem and Nori's Connectivity Theorem. Next, we construct in the
limit case explicit families of higher Chow groups which span the non trivial
cohomology classes in $X$. We give examples of indecomposable classes. The
construction suggests a conjecture predicting that in the limit case the
cokernel of the restriction map $H^k(Y\times B)\to H^k(X)$ should always be
algebraic, containing Nori's Connectivity Theorem and our previous work on the
Noether-Lefschetz locus as special cases.
