Hodge theory and K-stability of some very symmetric hypersurfaces

We study some interesting hypersurfaces that naturally arise when studying the period map on the moduli space of hypersurfaces, in the context of Sung Gi Park’s recent work on studying the GIT moduli space of hypersurfaces via the minimal exponent. We compute the Hodge structure on the singular cohomology and the intersection cohomology of these hypersurfaces, and also show the $K$-polystability of certain mildly singular degenerate hypersurfaces among them. In particular, the following hypersurface is $K$-polystable for $l\ge 2$: $$\{x_{11}\cdots x_{1d}+\dots +x_{ld}\cdots x_{ld}=0\}\subset {\text{PP}}^{ld-1}\mathrm{.}$$