On the Chow ring of certain hypersurfaces in a Grassmannian

Abstract

This small note is about Plücker hyperplane sections $X$ of the Grassmannian $\mathrm{Gr}(3,V_{10})$. Inspired by the analogy with cubic fourfolds, we prove that the only non-trivial Chow group of $X$ is generated by Grassmannians of type $\mathrm{Gr}(3,W_6)$ 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.

arXiv comment11 pages, to appear in Le Matematiche, comments welcome