On Teissier's example of an equisingularity class that cannot be defined over the rationals
Adam Parusiński, Laurentiu Paunescu
TL;DR
This work revisits Teissier's classical idea that a cone over Grünbaum's real-line arrangement, upon complexification, yields a surface singularity not Whitney equisingular to any singularity defined over $\mathbb{Q}$. It corrects Grünbaum's arrangement to admit a rational equation and provides a self-contained Whitney-theoretic proof that the tangent-cone data cannot be defined over $\mathbb{Q}$, avoiding the flawed LT79 argument. The authors prove a robust Main Theorem establishing absence of exceptional tangents and a Whitney stratification for the normal cone under suitable hypotheses, and they also analyze the limits of deformation-to-tangent-cone equisingularity, presenting a concrete counterexample. Altogether, the paper clarifies the arithmetic constraints on equisingularity, refines when deformation-to-the-normal-cone can be equisingular, and highlights the precise boundary between rational definability and geometric singularity behavior in Teissier's construction.
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.