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arXiv · 2508.04439

A minimal resolution for the Jacobian ideal of a generic curve arrangement

Alexandru Dimca·Gabriel Sticlaru
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Abstract

We consider a nodal curve $C$ in the complex projective plane whose irreducible components $C_i$ are smooth. A minimal set of generators $G$ for the first and second syzygy modules of the Jacobian ideal of $C$ are described, using recent results by Th. Kahle, H. Schenck, B. Sturmfels and M. Wiesmann on the likelihood correspondence. The elements of $G$ have explicit formulas in terms of the equations $f_i=0$ of the irreducible components $C_i$ of $C$. Similar results, including extensions to hypersurfaces arrangements in $\mathbb{P}^n$ were obtained by R. Burity, Z. Ramos, A. Simis and St. Toh\u aneanu with a genericity assumption which may not be easy to test in practice.

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Alexandru Dimca, Gabriel Sticlaru. 2025-08-06. A minimal resolution for the Jacobian ideal of a generic curve arrangement. https://arxiv.org/abs/2508.04439

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