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

## TL;DR

This paper investigates the behavior of algebraic cycles on hyperplane sections of specific hypersurfaces in projective space, focusing on the non-injectivity of push-forward maps in Chow groups for very general sections.

## Contribution

It provides new insights into the non-injectivity of push-forward homomorphisms on Chow groups for hyperplane sections of certain hypersurfaces in projective space.

## Key findings

- Identification of non-injectivity loci for specific hypersurfaces
- Analysis of Chow group push-forward homomorphisms
- Results applicable to very general hyperplane sections

## 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.