Quasi-homogeneous singularities of projective hypersurfaces and Jacobian syzygies

TL;DR

The paper establishes a practical criterion linking Jacobian syzygies to quasi-homogeneous singularities of hypersurfaces with isolated singularities: a singular point $p$ is quasi-homogeneous iff there exists a Jacobian syzygy with a nonvanishing coordinate at $p$, equivalently ${\rm rk}\,M_f(p)\ge1$ for the first syzygy matrix $M_f$. It extends this criterion to a geometric plane-curve setting, showing $\mu(C)=\tau(C)$ is equivalent to the irreducibility of the associated scheme $Z_f$ and to all singularities being quasi-homogeneous, with an explicit Hilbert-Burch description of $Z_f$ and a precise class formula $Z_f\equiv \alpha h_1^2+\beta h_1h_2+\gamma h_2^2$ where $\alpha=(d-1)^2-\tau(C)$, $\beta=d-1$, $\gamma=1$. The work connects local singularity types to global Jacobian data via the symmetric algebra $S(J_f)$, the polar map, and the Cohen–Macaulay properties of related schemes, providing computationally accessible tests and highlighting deeper links to Rees algebras, the Koszul hull, and potential residual structures. These results offer a robust framework to identify quasi-homogeneous singularities and to study the geometry of hypersurfaces through Jacobian syzygies.

Abstract

We prove an unexpected general relation between the Jacobian syzygies of a projective hypersurface $V\subset \mathbb{P}^n$ with only isolated singularities and the nature of its singularities. This allows to establish a new method for the identification of quasi-homogeneous hypersurface isolated singularities. The result gives an insight on how the geometry is reflected in the Jacobian syzygies and extends previous results of the first, second and last author for free and nearly free plane curves [1].

Theorems & Definitions (35)

• Theorem 1.1
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Example 2.4
• Theorem 3.1
• Corollary 3.2
• Theorem 3.3
• Corollary 3.4
• Proposition 3.5