Asymptotic bounds for Nori's connectivity theorem

Let $Y$ be a smooth complex projective variety. I study the cohomology of smooth families of hypersurfaces $\mathcal{X}\to B$ for $B\subset\mathbb{P}\mathrm{H}^0(Y,\mathcal{O}(d))$ a codimension $c$ subvariety. I give an asymptotically optimal bound on $c$ and $k$ when $d\to\infty$ for the space $\mathrm{H}^k(Y\times B,\mathcal{X},\mathbb{Q} )$ to vanish, thus extending the validity of the Lefschetz Hyperplane Section Theorem and Nori's Connectivity Theorem (1993). Next, I construct in the limit case explicit families of higher Chow groups whose class does not vanish in $\mathrm{H}^k(Y\times B,\mathcal{X},\mathbb{Q} )$. Some of them are indecomposable. This suggests that in the limit case the space $\mathrm{H}^k(Y\times B,\mathcal{X},\mathbb{Q} )$ should be spanned by higher Chow groups, containing Nori's and Otwinowska's results as special cases.