Syzygies of Jacobian ideals and weighted homogeneous singularities

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.

version 2: presentation of results slightly modified due to the addition a key reference by L. Busé and J.-P. Jouanolou

References in corpus (3)

• Generalization of theorems of Griffiths and Steenbrink to hypersurfaces with ordinary double points
• Syzygies of Jacobian ideals and defects of linear systems
• Homology of Homogeneous Divisors