Addition–deletion results for plus-one generated curves

Abstract

In the recent paper A. Dimca proves that when one adds to or deletes a line from a free curve, the resulting curve is either free or plus-one generated. We prove the converse statements, we give an additional insight into the original deletion result, and we derive a characterization of free curves in terms of behavior to addition/deletion of lines. Incidentally we generalize a result on conic-line arrangements by H. Schenck and Ş. Tohăneanu that describes when the addition or the deletion of a projective line from a free curve results in a free curve. We catalogue the possible splitting types of the bundle of logarithmic vector fields associated to a plus-one generated curve.