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

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 $n\le 8$, describe models for each $n$ explicitly and prove that sextic double solids with $n>3$ are birationally non-rigid.

52 pages, to appear in Nagoya Mathematical Journal. The proof of Theorem A in Section 3 was split into subsections for clarity. Match equation and table numbering with journal version