Birational rigidity of quartic three-folds with a double point of rank 3

TL;DR

This paper proves that a general quartic threefold with a specific double point singularity is birationally rigid, with a detailed structure of its birational automorphism group, contributing to the classification of such algebraic varieties.

Contribution

It establishes the birational rigidity of quartic threefolds with a rank 3 double point singularity and describes their birational automorphism group structure.

Findings

The variety is birationally rigid.

The birational automorphism group is a free product of 25 cyclic groups of order 2.

The set of non-rigid quartics has codimension at least 3.

Abstract

We prove that a general three-dimensional quartic $V$ in the complex projective space ${\mathbb P}^4$, the only singularity of which is a double point of rank 3, is a birationally rigid variety. Its group of birational self-maps is, up to the finite subgroup of biregular automorphisms, a free product of 25 cyclic groups of order 2. It follows that the complement to the set of birationally rigid factorial quartics with terminal singularities is of codimension at least 3 in the natural parameter space.