A Syzygy Rank Characterization of Strongly Euler Homogeneity for Projective Hypersurfaces

2025/10/06 by Liao, Xia, Zhang, Xiping
#14B05 #14C17 #32S60 #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2510.04482

Abstract

In this paper we give a characterization of strongly Euler homogeneous singular points on a reduced complex projective hypersurface D=V(f)⊂ \PPⁿ 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.

Citations

• Logrithmic Versions of Ginzburg's Sharp Operation for Free Divisors
• Quasi-homogeneous singularities of projective hypersurfaces and Jacobian syzygies
• A characterization of quasi-homogeneous singularities of free and nearly free plane curves

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