On plane conic arrangements with nodes and tacnodes

TL;DR

This paper investigates arrangements of smooth plane conics with nodes and tacnodes, providing bounds on their singularities and analyzing their geometric properties like freeness, improving existing bounds for large arrangements.

Contribution

It introduces a new estimation for the number of singularities based on the number of conics and offers improved bounds on tacnodes, along with a detailed study of freeness.

Findings

Derived a linear bound on nodes and tacnodes in conic arrangements.

Established a new upper bound on tacnodes surpassing Miyaoka's bound for large conic numbers.

Analyzed conditions for freeness and nearly freeness in these arrangements.

Abstract

In the present paper, we study arrangements of smooth plane conics having only nodes and tacnodes as the singularities. We provide an interesting estimation on the number of nodes and tacnodes that depends only on a linear function of the number of conics. Based on that result, we obtain a new upper bound on the number of tacnodes which turns out to be better than Miyaoka's bound for a large enough number of conics. We also study the freeness and nearly freeness of such arrangements providing a detailed description.