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.
Categories: math.AG