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

TL;DR

This paper investigates the combinatorial properties and existence constraints of $\\mathscr{M}$-arrangements of conics and lines with ordinary singularities, focusing on arrangements with a single conic and multiple lines.

Contribution

It introduces new numerical constraints and detailed analysis for $\\mathscr{M}$-arrangements, especially those with one conic and lines, expanding understanding of their combinatorial structure.

Findings

Numerical constraints on the existence of $\\mathscr{M}$-arrangements with ordinary singularities.

Detailed analysis of arrangements consisting of lines and one conic.

Insights into the combinatorial limitations of such arrangements.

Abstract

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.