Algebraic cycles on hyperplane sections of hypersurfaces in ${\mathbb{P}}^n$ for $n=5,6$

Abstract

Let $X$ be a cubic hypersurface in ${\mathbb{P}}^6$ or a hypersurface of degree greater than equal to $7$ in ${\mathbb{P}}^5$. In this note we try to understand, for a very general hyperplane section of $X$, the non-injectivity locus of the corresponding push-forward homomorphism at the level of Chow group of certain dimension.

9 pages, comments are welcome