Factoriality and birational rigidity of two families of singular quartic three-folds

TL;DR

This paper investigates two specific families of singular quartic three-folds in complex projective space, establishing their factoriality, birational rigidity, and describing their birational automorphism groups.

Contribution

It proves factoriality and birational rigidity for general members of these two families of singular quartic three-folds, a novel result in the classification of such varieties.

Findings

General quartics are factorial and birationally rigid.

The birational automorphism groups are explicitly described.

The families have codimension 3 in the parameter space.

Abstract

In this paper we study two families of three-dimensional quartics in the complex projective space ${\mathbb P}^4$: hypersurfaces with a unique quadratic singularity of rank 3, which is resolved by two blowups, and hypersurfaces with two quadratic singularities of rank 3 and 4, respectively. Both families have codimension 3 in the natural parameter space. For a Zariski general quartic in each of these families we prove factoriality and birational rigidity and describe its group of birational self-maps.