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

Abstract: In a paper, after introducing the notion of plus-one generated hyperplane arrangements, Takuro Abe has shown that if we add (resp. delete) a line to (resp. from) a free line arrangement, then the resulting line arrangement is either free or plus-one generated. In this note we prove that the same properties hold when we replace the line arrangement by a free curve and add (resp. delete) a line. The proof uses a new version of a key result due originally to H. Schenck, H. Terao and M. Yoshinaga, in which no quasi homogeneity assumption is needed. Two conjectures about the Tjurina number of a union of two plane curve singularities are also stated. As a geometric application, we show that, under a mild numerical condition, the projective closure of a contractible, irreducible affine plane curve is either free or plus-one generated, using a deep result due to U. Walther.