On Teissier's example of an equisingularity class that cannot be defined over the rationals

TL;DR

This paper corrects and completes Teissier's example demonstrating a surface singularity that cannot be defined over the rationals, highlighting subtle issues in equisingularity and field definability.

Contribution

It provides a corrected version of Teissier's example and a comprehensive proof of the non-rational definability of a specific surface singularity.

Findings

The corrected example confirms the existence of singularities not definable over Q.

A complete proof of Teissier's original result is provided.

The work clarifies the relationship between equisingularity and field of definition.

Abstract

A result of Teissier says that the cone over one of classical polygon examples in the real projective space gives, by complexification, a surface singularity which is not Whitney equisingular to a singularity defined over the field of rational numbers Q. In this note we correct the example and give a complete proof of Tesissier's result.