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The surface parametrizing cuboids

Michael Stoll, Damiano Testa

TL;DR

This work studies the surface $\bar{S}$ parametrizing cuboids and its minimal desingularization $S$, connecting the classical rational box problem to the arithmetic of $S$. It provides a full determination of the automorphism group $\operatorname{Aut}(S)$ and the geometric Picard group $\operatorname{Pic} S$ as a Galois module, proving $\operatorname{Aut}(S)$ has order $1536$ and that $\operatorname{Pic} S$ is a free lattice of rank $64$ with discriminant $-2^{28}$, generated by explicit curves and the $48$ exceptional divisors. By combining the Galois action with the automorphism group, the authors show $\operatorname{Pic} S$ is saturated and that the algebraic Brauer–Manin obstruction vanishes, enabling a detailed study of curves on $\bar{S}$. They place $\bar{S}$ in a modular framework via a quotient from $X\times X$ and related Weil restrictions, yielding a rich set of genus-$5$ fibrations and K3 quotients; this modular view, together with the Picard data, allows a near-complete classification of low-degree curves on $\bar{S}$ and sheds light on the rational box problem and weak approximation on $S$.

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 $\bar{S}$.

The surface parametrizing cuboids

TL;DR

This work studies the surface parametrizing cuboids and its minimal desingularization , connecting the classical rational box problem to the arithmetic of . It provides a full determination of the automorphism group and the geometric Picard group as a Galois module, proving has order and that is a free lattice of rank with discriminant , generated by explicit curves and the exceptional divisors. By combining the Galois action with the automorphism group, the authors show is saturated and that the algebraic Brauer–Manin obstruction vanishes, enabling a detailed study of curves on . They place in a modular framework via a quotient from and related Weil restrictions, yielding a rich set of genus- fibrations and K3 quotients; this modular view, together with the Picard data, allows a near-complete classification of low-degree curves on and sheds light on the rational box problem and weak approximation on .

Abstract

We study the surface 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 . Let be the minimal desingularization of (which has 48 isolated singular points). The main result of this paper is the explicit determination of the Picard group of , including its structure as a Galois module over . 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 (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 , to study curves of low degree on . In this way, we completely classify all integral curves of degree at most 6 on .

Paper Structure

This paper contains 7 sections, 19 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. The Automorphism Group
  3. The Picard Group
  4. The cuboid surface as a modular surface
  5. Fibrations in curves of genus 5
  6. K3 quotients
  7. Curves of low degree on $\bar{S}$

Key Result

Theorem 1

$\operatorname{Aut}_{\overline{\mathbb{Q}}}(S) = \operatorname{Aut}_{\overline{\mathbb{Q}}}(\bar{S})$ is an explicitly given group $G$ of order 1536. Its action on $\bar{S}$ extends to a linear action on the ambient ${\mathbb P}^6$.

Theorems & Definitions (33)

  • Theorem 1
  • Theorem 2
  • Lemma 3
  • proof
  • Proposition 4
  • proof
  • Lemma 5
  • proof
  • Definition 6
  • Proposition 7
  • ...and 23 more