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

• Published in: Bulletin of mathematical sciences Vol. 14; no. 3
• Main Author: Dimca, Alexandru
• Format: Journal Article
• Language: English
• Published: World Scientific Publishing Company 01.12.2024; World Scientific Publishing
• Subjects: Free curve; plus-one generated curve; Free curve; plus-one generated curve
• ISSN: 1664-3607; 1664-3615
• DOI: 10.1142/S1664360724500073

In a recent paper, after introducing the notion of plus-one generated hyperplane arrangements, Takuro Abe has shown that if we add (respectively, delete) a line to (respectively, from) a free line arrangement, then the resulting line arrangement is either free or plus-one generated. In this paper, we prove that the same properties hold when we replace the line arrangement by a free curve and add (respectively, 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.