The surface parametrizing cuboids

TL;DR

This paper investigates the algebraic surface parametrizing cuboids, determines its Picard group explicitly, and classifies low-degree integral curves, contributing to the understanding of rational solutions related to the existence of rational cuboids.

Contribution

It explicitly computes the Picard group of the surface parametrizing cuboids and classifies all low-degree integral curves on it, advancing the algebraic understanding of rational cuboids.

Findings

Determined the Picard group of the desingularized surface.

Identified the structure of the Picard group as a Galois module.

Classified all integral curves of degree at most 6 on the surface.

Abstract

We study the surface $\bar{S}$ parametrizing cuboids: it is defined by the equations relating the sides, face diagonals and long diagonal of a rectangular box. It is an open problem whether a `rational box' exists, i.e., a rectangular box all of whose sides, face diagonals and long diagonal have (positive) rational length. The question is equivalent to the existence of nontrivial rational points on $\bar{S}$. Let $S$ be the minimal desingularization of $\bar{S}$ (which has 48 isolated singular points). The main result of this paper is the explicit determination of the Picard group of $S$, including its structure as a Galois module over $\mathbb Q$. The main ingredient for showing that the known subgroup is actually the full Picard group is the use of the combined action of the Galois group and the geometric automorphism group of $S$ (which we also determine) on the Picard group. This reduces the proof to checking that the hyperplane section is not divisible by 2 in the Picard group. We use our explicit knowledge of the Picard group, together with that of a K3 surface obtained as a quotient of $S$, to study curves of low degree on $\bar{S}$. In this way, we completely classify all integral curves of degree at most 6 on $\bar{S}$.