Birational rigidity and Q-factoriality of a singular double quadric

TL;DR

This paper establishes the birational rigidity and automorphism group of certain singular double quadrics, and determines conditions for Q-factoriality related to the number of singularities.

Contribution

It proves birational rigidity and computes automorphisms of a class of singular double quadrics, also identifying the threshold for Q-factoriality based on singularity count.

Findings

X is Q-factorial with up to 11 singularities

X is not Q-factorial with 12 singularities

Automorphism group of X is explicitly calculated

Abstract

We prove birational rigidity and calculate the group of birational automorphisms of a nodal Q-factorial double cover $X$ of a smooth three-dimensional quadric branched over a quartic section. We also prove that $X$ is Q-factorial provided that it has at most 11 singularities; moreover, we give an example of a non-Q-factorial variety of this type with 12 simple double singularities.