Q-factorial quartic threefolds

TL;DR

This paper establishes conditions under which nodal quartic threefolds are Q-factorial, specifically when they have up to 12 singular points and do not contain planes, with a special case exception.

Contribution

It provides a new criterion for Q-factoriality of nodal quartic threefolds based on singularity count and geometric properties, including a detailed analysis of a special quartic with 12 singularities.

Findings

Q-factoriality holds for nodal quartic threefolds with ≤12 singular points without planes.

A unique quartic with 12 singularities containing a quadric surface is an exception.

Geometric constructions related to the exceptional quartic are presented.

Abstract

We prove that a nodal quartic threefold $X$ containing no planes is $Q$-factorial provided that it has not more than 12 singular points, with the exception of a quartic with exactly 12 singularities containing a quadric surface. We give some geometrical constructions related to the latter quartic.