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)⊂ ⁿ using the Jacobian syzygies of f. The characterization compares the ranks of the first syzygy matrices of the global Jacobian ideal Jf and its quotient Jf/(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.