Syzygies of Jacobian ideals and defects of linear systems

TL;DR

This paper explores the connection between syzygies of Jacobian ideals of homogeneous polynomials and the defects in linear systems associated with the singular locus of hypersurfaces with isolated singularities.

Contribution

It establishes a new relation between syzygies of Jacobian ideals and the defect of linear systems on hypersurfaces with isolated singularities.

Findings

Describes the relation between syzygies and defects of linear systems.

Applicable to hypersurfaces with only isolated singularities.

Provides a framework linking algebraic syzygies to geometric defects.

Abstract

Our main result describes the relation between the syzygies involving the first order partial derivatives $f_0,...,f_n$ of a homogeneous polynomial $f\in \C[x_0,...x_n]$ and the defect of the linear systems vanishing on the singular locus subscheme $\Sigma_f=V(f_0,...,f_n)$ of the hypersurface $D:f=0$ in the complex projective space $\PP^n$, when $D$ has only isolated singularities.