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.

18 pages. Comments are welcome. v5: final version

References in corpus (4)

• Openness of uniform K-stability in families of $\mathbb{Q}$-Fano varieties
• Birational rigidity is not an open property
• K-stability of birationally superrigid Fano varieties
• Higher Codimensional Alpha Invariants and Characterization of Projective Spaces

Cited by in corpus (2)

• Higher Codimensional Alpha Invariants and Characterization of Projective Spaces
• K-stability of birationally superrigid Fano 3-fold weighted hypersurfaces