Birational superrigidity is not a locally closed property

Abstract

We prove an optimal result on the birational rigidity and K-stability of index $1$ hypersurfaces in $\mathbb{P}^{n+1}$ with ordinary singularities when $n\gg 0$ and also study the birational superrigidity and K-stability of certain weighted complete intersections. As an application, we show that birational superrigidity is not a locally closed property in moduli. We also prove (in the appendix) that the alpha invariant function is constructible in some families of complete intersections.