Year: 2026
Type: article
Abstract: We study the surface S ¯ \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 cuboid’ 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 non-trivial rational points on S ¯ \bar {S} . The surface S ¯ \bar S has 48 isolated singular points. Let S S be the minimal desingularization of S ¯ \bar S . The main result of this paper is the explicit determination of the Picard group of S S , including its structure as a Galois module over Q \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 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 S , to study curves of low degree on S ¯ \bar {S} . In this way, we completely classify all integral curves of degree at most 6 on S ¯ \bar {S} .
Source: Mathematics of Computation
DOI: 10.1090/mcom/4238
Authors: Michael Stoll, Damiano Testa
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: University of Warwick
OpenAlex ID: W7161063686