Saturation of Jacobian ideals: some applications to nearly free curves, line arrangements and rational cuspidal plane curves

TL;DR

This paper investigates the structure of the saturation of Jacobian ideals in nearly free plane curves, providing minimal resolutions, bounds on generators, and applications to rational cuspidal curves and line arrangements.

Contribution

It presents the minimal resolution of the saturated Jacobian ideal for nearly free curves and explores its applications to specific algebraic and geometric configurations.

Findings

The saturated Jacobian ideal can be generated by at most 4 polynomials.

Provides explicit minimal resolutions for these ideals.

Applications include insights into rational cuspidal curves and line arrangements.

Abstract

In this note we describe the minimal resolution of the ideal $I_f$, the saturation of the Jacobian ideal of a nearly free plane curve $C:f=0$. In particular, it follows that this ideal $I_f$ can be generated by at most 4 polynomials. Related general results by Hassanzadeh and Simis on the saturation of codimension 2 ideals are discussed in detail. Some applications to rational cuspidal plane curves and to line arrangements are also given.