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

TL;DR

This paper provides explicit minimal generators for the syzygy modules of the Jacobian ideal of a nodal curve arrangement, extending known results to more general hypersurface arrangements with practical formulas.

Contribution

It introduces explicit formulas for minimal generators of the Jacobian ideal's syzygies for generic nodal curves, generalizing previous work to hypersurfaces in projective space.

Findings

Explicit formulas for generators of the Jacobian ideal's syzygies.

Extension of results to hypersurface arrangements in projective space.

Comparison with prior genericity assumptions in the literature.

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.