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Let $X$ be a cubic hypersurface in $\mathbb P^6$ or a hypersurface of degreegreater 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 thecorresponding push-forward homomorphism at the level of Chow group of certaindimension.

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