On the Structure of Second Jacobian Ideals

TL;DR

This paper investigates the algebraic structure of second Jacobian ideals of hypersurfaces, providing a decomposition that reveals their properties and applications to invariants of hypersurface singularities.

Contribution

It introduces a novel decomposition of the second Jacobian ideal and proves the second Nash blow-up algebra is a contact invariant for hypersurface singularities.

Findings

Decomposition of second Jacobian ideal into factors.

Power of Jacobian ideal relates to the ideal's structure.

Second Nash blow-up algebra is a contact invariant.

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.