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

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.

💡 Research Summary

The paper revisits a claim made by Bernard Teissier in 1990 that the complexification of the cone over a classical polygonal line arrangement in real projective space yields a surface singularity which is not Whitney‑equisingular to any surface singularity defined over the rational numbers ℚ. The authors identify two flaws in Teissier’s original argument and provide a corrected, complete proof of the result.

First, the authors examine the Grünbaum arrangements of nine lines (denoted C and C′) built from a regular pentagon. Teissier’s example implicitly uses the arrangement C′, but the original exposition did not rigorously show that this configuration cannot be defined over ℚ. By adding an extra line HE to C′, the authors obtain a ten‑line arrangement (still called C′) whose defining polynomial is the product of ten linear forms. They analyze the action of the Galois group G of the field obtained by adjoining the real coordinates of the fifth roots of unity to ℚ. While the product of the first nine linear forms is G‑invariant (hence has rational coefficients), the inclusion of the tenth line destroys G‑invariance. Using a careful weight‑counting argument on intersection points, they prove that the full product cannot be fixed by G, which implies that the arrangement cannot be described by equations with rational coefficients. This corrects the combinatorial part of Teissier’s example.

Second, Teissier’s proof relied on a theorem (Theorem 2.1.1 in