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In a recent paper, after introducing the notion of plus-one generatedhyperplane arrangements, Takuro Abe has shown that if we add (resp. delete) aline to (resp. from) a free line arrangement, then the resulting linearrangement is either free or plus-one generated. In this note we prove thatthe same properties hold when we replace the line arrangement by a free curveand add (resp. delete) a line. The proof uses a new version of a key result dueoriginally to H. Schenck, H. Terao and M. Yoshinaga, in which no quasihomogeneity assumption is needed. Two conjectures about the Tjurina number of aunion of two plane curve singularities are also stated. As a geometricapplication, we show that, under a mild numerical condition, the projectiveclosure of a contractible, irreducible affine plane curve is either free orplus-one generated, using a deep result due to U. Walther.

On free and plus-one generated curves arising from free curves by addition-deletion of a line