Birational geometry of sextic double solids with a compound $A_n$ singularity

TL;DR

This paper investigates the birational geometry of sextic double solids with a specific type of singularity, establishing bounds on the singularity and demonstrating rigidity or non-rigidity depending on the singularity's complexity.

Contribution

It provides a sharp bound on the compound $A_n$ singularity degree and characterizes the birational rigidity of these solids based on the singularity.

Findings

Bound $n \,\leq\, 8$ for the singularity degree.

Explicit models for each $n$ up to 8.

Sextic double solids with $n > 3$ are birationally non-rigid.

Abstract

Sextic double solids, double covers of $\mathbb P^3$ branched along a sextic surface, are the lowest degree Gorenstein Fano 3-folds, hence are expected to behave very rigidly in terms of birational geometry. Smooth sextic double solids, and those which are $\mathbb Q$-factorial with ordinary double points, are known to be birationally rigid. In this article, we study sextic double solids with an isolated compound $A_n$ singularity. We prove a sharp bound $n \leq 8$, describe models for each $n$ explicitly and prove that sextic double solids with $n > 3$ are birationally non-rigid.