A Syzygy Rank Characterization of Strongly Euler Homogeneity for Projective Hypersurfaces
Xia Liao, Xiping Zhang
TL;DR
The paper develops a syzygy-rank framework to characterize strongly Euler homogeneous singularities on projective hypersurfaces. By translating Rodríguez's local criterion into global algebra via the first syzygy matrix $M'_f$ of the Jacobian ideal $J_f$ and its augmented version $M_f$ for $J_f/(f)$, it yields a pointwise condition $\operatorname{rk} M'_f(p)=\operatorname{rk} M_f(p)$ indicating strong Euler homogeneity. The work connects to log characteristic cycles, showing geometric criteria involving $Z_f$, $S_f$, and the incidence $I$, and extends the theory to toric varieties with a logarithmic defect $\operatorname{Def}_{X,f}(p)$ that accounts for ambient geometry. Collectively, these results refine prior characterizations (notably ABDM25/ABM25) and provide robust algebraic tools for detecting strong Euler homogeneity across broad classes of projective varieties.
Abstract
In this paper we give a characterization of strongly Euler homogeneous singular points on a reduced complex projective hypersurface $D=V(f)\subset \PP^n$ 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 $J_f/(f)$. 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.