On the locus of non-rigid hypersurfaces

TL;DR

This paper investigates the geometric properties of hypersurfaces in projective space, establishing a lower bound on the codimension of those lacking factoriality or birational superrigidity.

Contribution

It provides a new lower bound on the codimension of hypersurfaces that are not factorial or not birationally superrigid in the parameter space.

Findings

The Zariski closure of hypersurfaces not factorial or not birationally superrigid has codimension at least inom{M-3}{2}+1.

The result applies to hypersurfaces of degree M in bP^M with M  5.

The work advances understanding of the distribution of special hypersurfaces in algebraic geometry.

Abstract

We show that the Zariski closure of the set of hypersurfaces of degree $M$ in ${\mathbb P}^{M}$, where $M\geq 5$, which are either not factorial or not birationally superrigid, is of codimension at least $\binom{M-3}{2}+1$ in the parameter space.