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

TL;DR

This paper extends the understanding of free and plus-one generated curves by showing that adding or deleting a line preserves these properties, generalizing previous results from line arrangements to curves.

Contribution

It proves that free and plus-one generated properties are maintained when adding or deleting a line from free curves, removing the quasi homogeneity restriction.

Findings

The properties hold for free curves when lines are added or removed.

A new version of a key result by Schenck, Terao, and Yoshinaga is established.

Under certain conditions, the projective closure of a contractible, irreducible affine plane curve is free or plus-one generated.

Abstract

In a recent 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.