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

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