Nori’s connectivity theorem compares the cohomology of $X\times B$ and $Y_B$, where $Y_B$ is any locally complete quasiprojective family of sufficiently ample complete intersections in $X$. When $X$ is the projective space, and we consider hypersurfaces of degree $d$, it is possible to give an explicit bound for $d$, sufficient to conclude that the Connectivity Theorem holds. We show that this bound is optimal, by constructing for lower $d$ classes on $Y_B$ not coming from the ambient space. As a byproduct we get the non-triviality of the higher Chow groups of generic hypersurfaces of degree $2n$ in $\mathbb{P}^{n+1}$.

AMS 2000 Mathematics subject classification: Primary 14C25; 14D07; 14J70