The surface parametrizing cuboids

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 $\bar{S}$.