Construction of free curves by adding lines to a given curve

TL;DR

This paper develops methods to construct free plane curves by adding specific lines to a given curve, introduces the concept of supersolvable curves, and provides new examples of maximizing curves, advancing understanding of free curve arrangements.

Contribution

It introduces a new construction technique for free curves, proposes the supersolvable curve conjecture, and presents novel examples of maximizing curves in degrees 8 and 9.

Findings

Construction of free curves by adding lines with specific intersection properties.

Evidence supporting the supersolvable curve conjecture.

New examples of maximizing curves in degrees 8 and 9.

Abstract

In the present note we construct new families of free plane curves starting from a curve $C$ and adding high order inflectional tangent lines of $C$, lines joining the singularities of the curve $C$, or lines in the tangent cone of some singularities of $C$. These lines $L$ have in common that the intersection $C \cap L$ consists of a small number of points. We introduce the notion of a supersolvable plane curve and conjecture that such curves are always free, as in the known case of line arrangements. Some evidence for this conjecture is given as well, both in terms of a general result in the case of quasi homogeneous singularities and in terms of specific examples. We construct a new example of maximizing curve in degree 8 and the first and unique known example of maximizing curve in degree 9. In the final section, we use a stronger version of a result due to Schenck, Terao and Yoshinaga to construct families of free conic-line arrangements by adding lines to the conic-line arrangements of maximal Tjurina number recently classified by V. Beorchia and R. M. Mir\'o-Roig in arXiv:2303.04665.