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 notation is a natural generalization of maximizing plane curves which are well-known in the theory of algebraic surfaces. We focus here on $\mathscr{M}$-arrangements of conics and lines with ordinary singularities of multiplicity less than five and we provide various numerical constraints on their existence, particularly in terms of their weak combinatorics. Moreover, we study in detail the scenario when our $\mathscr{M}$-arrangements consist of lines and just one conic.