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

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.