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On the Structure of Second Jacobian Ideals

Fei Ye

TL;DR

The paper investigates the second Jacobian ideal of a hypersurface and proves a precise decomposition: $\mathcal{J}_2(F)=\mathcal{J}_1(F)^{n+1}+\mathcal{J}_1(F)^{n-2}\mathcal{Q}(F)$. This is established via a detailed analysis of the second Jacobian matrix $\operatorname{Jac}_2(F)$, exploiting block-structural reductions of its square submatrices and a case-by-case examination of maximal minors, showing determinants lie in the desired products of Jacobian ideals and $\mathcal{Q}(F)$. The result yields a robust link between higher Jacobian structures and the classical Jacobian ideal, enabling an elementary proof that the second Nash blow-up algebra is a contact invariant for hypersurface singularities. As an application, the paper derives a concrete description of the invariant behavior under isomorphisms and illustrates the decomposition with explicit cases, including a corroborating example $F=x^3-y^2$. Overall, the work connects the algebraic structure of $\mathcal{J}_2(F)$ to Nash blow-ups and Fitting-ideal frameworks, enriching the understanding of singularities and their local invariants.

Abstract

We show that the second Jacobian ideal of a hypersurface can be decomposed such that a power of the Jacobian ideal becomes a factor. As an application of the decomposition, we present an elementary proof establishing that the second Nash blow-up algebra of a hypersurface singularity is a contact invariant.

On the Structure of Second Jacobian Ideals

TL;DR

The paper investigates the second Jacobian ideal of a hypersurface and proves a precise decomposition: . This is established via a detailed analysis of the second Jacobian matrix , exploiting block-structural reductions of its square submatrices and a case-by-case examination of maximal minors, showing determinants lie in the desired products of Jacobian ideals and . The result yields a robust link between higher Jacobian structures and the classical Jacobian ideal, enabling an elementary proof that the second Nash blow-up algebra is a contact invariant for hypersurface singularities. As an application, the paper derives a concrete description of the invariant behavior under isomorphisms and illustrates the decomposition with explicit cases, including a corroborating example . Overall, the work connects the algebraic structure of to Nash blow-ups and Fitting-ideal frameworks, enriching the understanding of singularities and their local invariants.

Abstract

We show that the second Jacobian ideal of a hypersurface can be decomposed such that a power of the Jacobian ideal becomes a factor. As an application of the decomposition, we present an elementary proof establishing that the second Nash blow-up algebra of a hypersurface singularity is a contact invariant.

Paper Structure

This paper contains 22 sections, 14 theorems, 137 equations.

Table of Contents

  1. Introduction
  2. second Jacobian matrices
  3. Step 1.
  4. Case 1.1
  5. Case 1.2
  6. Step 2.
  7. Case 2.1
  8. Case 2.2
  9. Some Results on Minors
  10. Decomposition of Second Jacobian Ideals
  11. Case 1:
  12. Case 2:
  13. Case 1: $c=2$.
  14. Case 2: $c=3$.
  15. Subcase 2.1: Suppose $i=j_a$ for some $1\le a \le n-2$.
  16. ...and 7 more sections

Key Result

Lemma 2.5

Let $S\subseteq \{1, \dots, n\}$ be a subset of cardinality $r$ and $M$ an $(r+1)$-by-$(r+1)$ submatrix of ${\operatornamewithlimits{Jac}}_2(F)$, where the set of column indices of $M$ is $S$ and the set of row indices of $M$ is $\{0\}\cup S$. Then the matrix $M$ is permutation equivalent to a block where $B_i$ are diagonal square matrices (could be $0$-dimensional) and $D$ is a square matrix obta

Theorems & Definitions (35)

  • Conjecture 1.1: Ramirez2024
  • Definition 2.1
  • Remark
  • Remark
  • Example 2.3
  • Example 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • ...and 25 more