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Reconstruction of a Hypersurface Singularity from its Moduli Algebra

João Hélder Olmedo Rodrigues

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.

Reconstruction of a Hypersurface Singularity from its Moduli Algebra

Abstract

In this paper we present a constructive method to characterize ideals of the local ring of germs of holomorphic functions at which arise as the moduli ideal , for some . 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.
Paper Structure (5 sections, 6 theorems, 24 equations)

This paper contains 5 sections, 6 theorems, 24 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Computation of $\Delta_1$
  4. Two properties of ideals in $\mathscr{O}_{\mathbb{C}^n,0}$
  5. Main Result

Key Result

Theorem 1

Let $(X_f,0)\subset (\mathbb{C}^n,0)$ and $(X_g,0)\subset (\mathbb{C}^n,0)$ denote two germs of complex hypersurfaces. The statements are equivalent:

Theorems & Definitions (39)

  • Theorem : Mather, Yau; Gaffney, Hauser
  • Remark 1.1
  • Example 1.3
  • Example 1.4
  • Remark 1.5
  • Example 1.6
  • Remark 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • ...and 29 more