The G\"opel variety

TL;DR

This paper rigorously proves the generators of the six-dimensional G"opel variety in projective space, correcting previous assumptions and clarifying the relations needed for its algebraic description.

Contribution

It establishes the precise generators of the G"opel variety, correcting earlier misconceptions about its defining relations.

Findings

The G"opel variety is generated by 120 linear, 35 cubic, and 35 quartic relations.

Previous claims about the G"opel variety being generated only by linear and cubic relations are false.

The paper provides detailed corrections to earlier results and clarifies the algebraic structure of the G"opel variety.

Abstract

In this paper we will prove that the six-dimensional G\"opel variety in $P^{134}$ is generated by 120 linear, 35 cubic and 35 quartic relations. This result was already obtained in [RS] , but the authors used a statement in [Co] saying that the G\"opel variety set theoretically is generated by the linear and cubic relations alone. Unfortunately this statement is false. There are 120 extra points. Nevertheless the results stated in [RS] are correct. There are required several changes that we will illustrate in some detail