# On the Chow ring of certain hypersurfaces in a Grassmannian

## TL;DR

This paper investigates the Chow ring structure of Plücker hyperplane sections of a Grassmannian, showing the main Chow group is generated by specific sub-Grassmannians and a subring injects into cohomology.

## Contribution

It establishes the generators of the Chow group for these hypersurfaces and proves an injection of a subring into cohomology, inspired by analogies with cubic fourfolds.

## Key findings

- The main Chow group is generated by Grassmannians of type Gr(3,W_6) contained in X.
- A certain subring of the Chow ring injects into cohomology.
- Provides new insights into the algebraic cycles of these hypersurfaces.

## Abstract

This small note is about Pl\"ucker hyperplane sections $X$ of the Grassmannian $\operatorname{Gr}(3,V_{10})$. Inspired by the analogy with cubic fourfolds, we prove that the only non-trivial Chow group of $X$ is generated by Grassmannians of type $\operatorname{Gr}(3,W_{6})$ contained in $X$. We also prove that a certain subring of the Chow ring of $X$ (containing all intersections of positive-codimensional subvarieties) injects into cohomology.