Reconstruction of a Hypersurface Singularity from its Moduli Algebra

TL;DR

This paper introduces a constructive method to identify ideals of local rings associated with hypersurface singularities, enabling the reconstruction of the singularity from its moduli algebra, solving a longstanding problem in singularity theory.

Contribution

It provides an effective, general solution to reconstruct hypersurface singularities from their moduli algebra, applicable to both isolated and non-isolated cases.

Findings

Developed a constructive characterization of ideals in local rings

Provided an effective method for hypersurface reconstruction

Solved the longstanding Reconstruction Problem from moduli algebra

Abstract

In this paper we present a constructive method to characterize ideals of the local ring $\mathscr{O}_{\mathbb{C}^n,0}$ of germs of holomorphic functions at $0\in\mathbb{C}^n$ which arise as the moduli ideal $\langle f,\mathfrak{m}\, j(f)\rangle$, for some $f\in\mathfrak{m}\subset\mathscr{O}_{\mathbb{C}^n,0}$. A consequence of our characterization is an effective solution to a problem dating back to the 1980's, called the Reconstruction Problem of the hypersurface singularity from its moduli algebra. Our results work regardless of whether the hypersurface singularity is isolated or not.