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

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. Tohăneanu with a genericity assumption which may not be easy to test in practice.