Construction of Free Curves by Adding Osculating Conics to a Given Cubic Curve.

The article focuses on constructing new families of free and nearly free plane curve arrangements by combining a plane cubic curve—specifically nodal cubics and the Fermat cubic—with their hyperosculating conics, which are conics having high-order contact at sextactic points of the cubic. It provides a complete characterization of free arrangements formed by a nodal cubic and up to three hyperosculating conics, showing that such arrangements are free or maximal Tjurina curves depending on the number of conics added. For the Fermat cubic, the 27 hyperosculating conics are partitioned into nine classes based on tangency properties, and the paper establishes criteria under which arrangements of the Fermat cubic with two or three hyperosculating conics are free or nearly free, supported by explicit computations and group action symmetries. The work relies on algebraic and computational tools, including the use of the polynomial ring over complex numbers, Jacobian syzygies, and symbolic computation software SINGULAR, to verify freeness conditions via Tjurina numbers and minimal degrees of Jacobian syzygies.