A Syzygy Rank Characterization of Strongly Euler Homogeneity for Projective Hypersurfaces

In this paper we give a characterization of strongly Euler homogeneous singular points on a reduced complex projective hypersurface $D=V(f)\subset {\text{PP}}^n$ using the Jacobian syzygies of $f$. The characterization compares the ranks of the first syzygy matrices of the global Jacobian ideal $J_f$ and its quotient $J_f\mathrm{/}(f)$. When $D$ has only isolated singularities, our characterization refines a recent result of Andrade-Beorchia-Dimca-Miró-Roig. We also prove a generalization of this characterization to smooth projective toric varieties.