Syzygies of Jacobian ideals and defects of linear systems

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