On $\mathscr{M}$-arrangements of conics and lines with ordinary singularities

In this paper we study combinatorial aspects of reduced plane curves known as $\mathscr{M}$-curves.

This notion is a natural generalization of maximizing plane curves, which are well-known in the theory of algebraic surfaces.

We focus on $\mathscr{M}$-arrangements of conics and lines with ordinary singularities of multiplicity at most four.

We provide numerical constraints on their existence, especially in terms of weak combinatorics, and then study in detail the case of arrangements consisting of one conic and lines.

We also construct a new example with one conic and eleven lines, prove boundedness results for real arrangements of this type, and record a regularity consequence for the associated Milnor algebra and module of Jacobian syzygies.