Which quartic double solids are rational?

TL;DR

This paper investigates the rationality of nodal quartic double solids, establishing conditions under which these three-dimensional algebraic varieties are rational or irrational based on their singular points.

Contribution

It provides a complete classification of the rationality of nodal quartic double solids depending on the number of singular points, filling a gap in algebraic geometry.

Findings

Nodal quartic double solids with up to six singular points are irrational.

Nodal quartic double solids with eleven or more singular points are rational.

The rationality depends critically on the number of singular points.

Abstract

We study the rationality problem for nodal quartic double solids. In particular, we prove that nodal quartic double solids with at most six singular points are irrational, and nodal quartic double solids with at least eleven singular points are rational.