Asymptotic bounds for Nori's connectivity theorem

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.