Quasi-homogeneous singularities of projective hypersurfaces and Jacobian syzygies

TL;DR

This paper establishes a new relation between Jacobian syzygies and the nature of isolated singularities in projective hypersurfaces, providing a novel method to identify quasi-homogeneous singularities and linking geometric properties with algebraic invariants.

Contribution

It introduces a new approach connecting Jacobian syzygies with the classification of hypersurface singularities, extending previous results to higher dimensions.

Findings

Relation between Jacobian syzygies and singularity types

New method for identifying quasi-homogeneous singularities

Extension of previous plane curve results to hypersurfaces

Abstract

We prove an unexpected general relation between the Jacobian syzygies of a projective hypersurface $V\subset \mathbb{P}^n$ with only isolated singularities and the nature of its singularities. This allows to establish a new method for the identification of quasi-homogeneous hypersurface isolated singularities. The result gives an insight on how the geometry is reflected in the Jacobian syzygies and extends previous results of the first, second and last author for free and nearly free plane curves [1].