Syzygies of Jacobian ideals and defects of linear systems

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.

version 4: the discussion of the case when the singular locus is a complete intersection is added. The a-invariant and the Castelnuovo-Mumford regularity of the Milnor algebra are also explored