Mather-Yau's type theorem for higher Nash blowup algebras

TL;DR

This paper extends the classical Mather-Yau theorem to higher Nash blowup algebras, showing they determine the local ring of hypersurface singularities over any field, thus generalizing and unifying previous results in singularity theory.

Contribution

It proves the stability of higher Nash blowup algebras under contact equivalence and establishes that these algebras fully determine the hypersurface singularity's local ring in a broad setting.

Findings

Higher Nash blowup algebras are invariant under contact equivalence.

The isomorphism type of the local ring is determined by higher Nash blowup algebras.

The stability conjecture for complex hypersurface singularities is confirmed.

Abstract

In this paper, we establish a Mather-Yau theorem for higher Nash blowup algebras, demonstrating that the isomorphism type of the local ring of any hypersurface singularity, defined over an arbitrary field, is fully determined by its higher Nash blowup algebras. The classical Mather-Yau theorem (1982) asserts that for isolated complex hypersurface singularities, the isomorphism type of the local ring is determined by the Tjurina algebra. In positive characteristic, this result was extended by considering the higher Tjurina algebras by Greuel and Pham (2017) under the assumptions of an algebraically closed ground field and isolated singularities. Our work begins by proving the stability of higher Nash blowup algebras under contact equivalence in a very general framework. Specifically, we show that the higher Nash blowup algebras of any system of elements in an analytic or geometric ring remain invariant under contact equivalence. For complex hypersurface singularities, this stability was conjectured by Hussain, Ma, Yau, and Zuo, and was recently verified by Le and Yasuda. Finally, the converse is established using a classical result of Samuel (1956).