Syzygies of Jacobian ideals and weighted homogeneous singularities

TL;DR

This paper characterizes weighted homogeneous singularities of projective hypersurfaces through the structure of Koszul syzygies among partial derivatives, providing a new criterion and computational improvements.

Contribution

It establishes a new criterion for weighted homogeneous singularities based on Koszul syzygies and enhances computational methods for listing Jacobian syzygies.

Findings

Weighted homogeneous singularities characterized by Koszul syzygies.

Answer to a question posed by Saito and the first author.

Improved algorithms for computing Jacobian syzygies in computer algebra systems.

Abstract

Let $V$ be a projective hypersurface having only isolated singularities. We show that these singularities are weighted homogeneous if and only if the Koszul syzygies among the partial derivatives of an equation for $V$ are exactly the syzygies with a generic first component vanishing on the singular locus subscheme of $V$. This yields in particular a positive answer in this setting to a question raised by Morihiko Saito and the first author. Finally we explain how our result can be used to improve the listing of Jacobian syzygies of a given degree by a computer algebra system such as Singular, CoCoA or Macaulay2.