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

TL;DR

The paper analyzes two codimension-3 families of quartic threefolds in $\mathbb P^4$ with isolated quadratic singularities, proving factoriality and birational rigidity for Zariski-general members. It combines cohomological vanishing arguments and a plane-section technique to establish factoriality (Pic $V_2$ generated by $H$, $Q_1$, $Q_2$) and to control maximal singularities via the Noether–Fano method. It then derives a precise description of the birational automorphism group as an extension $1\to B(V)\to Bir\ V\to Aut\ V\to 1$, with $B(V)$ generated by involutions linked to lines through the singular points, and shows that for general quartics $Aut\ V$ is finite or trivial. Together, these results advance the birational geography of three-dimensional quartics and support rigidity phenomena in codimension-3 strata.

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.