On a question of H. A. Schwarz

TL;DR

This paper investigates the rationality of a hyperelliptic variety related to tetrahedra with rational sides and volume, using Weddle surfaces, and suggests the variety is likely not rational over the rationals.

Contribution

The paper reexamines Schwarz's question on the rationality of the hyperelliptic variety using Weddle surfaces, providing new insights and partial evidence.

Findings

Otto Schulz proved the variety is rational over a7(a7a7a7a7)

The paper suggests the variety is likely not rational over a7

The approach uses the theory of Weddle surfaces

Abstract

This paper is a contribution to the study of tetrahedra with rational side lengths and rational volume. The problem can be formulated as a study of a hyperelliptic variety attached to the Cayley-Menger determinant. The problem posed by Schwarz was as to whether this variety is rational over the rationals. In a thesis that was never published Otto Schulz proved in 1912 that the variety was rational over $\mathbb Q(\sqrt{-1})$. This does not seem to have appeared eleswhere nor, apparently, has the question been studied since then. In this paper we take up an idea of Kurt Heegner to reprove the theorem using the theory of Weddle surfaces. Although this does not lead to a final answer to Schwarz question it seems likely that it is negative.