# Birational superrigidity is not a locally closed property

## TL;DR

This paper investigates the properties of birational superrigidity and K-stability in hypersurfaces and complete intersections, demonstrating that superrigidity is not a locally closed property in moduli spaces.

## Contribution

It establishes optimal conditions for birational rigidity and K-stability of hypersurfaces with singularities and shows that superrigidity is not locally closed in moduli.

## Key findings

- Birational superrigidity is not a locally closed property in moduli.
- Optimal results on birational rigidity and K-stability for hypersurfaces with ordinary singularities.
- Constructibility of the alpha invariant in certain families of complete intersections.

## 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.