Pencils of Conic-Line Curves

TL;DR

This paper investigates the maximum number of conic-line curves in special pencils, establishing bounds related to concurrent lines, and constructs examples with exactly four conic-line curves, including general position cases.

Contribution

It provides new bounds on the number of conic-line curves in pencils and constructs explicit families with a fixed number of such curves.

Findings

Upper bounds on conic-line curves related to concurrent lines

Construction of pencils with exactly four conic-line curves

Analysis of conic-line curves in general position

Abstract

In this paper, we study the restrictions on the number $m$ of conic-line curves in special pencils. The most general result we obtain is the relation between upper bounds on $m$ and the number $p$ of concurrent lines in these pencils. We construct a one-parameter family of pencils such that each pencil in the family contains exactly 4 conic-line curves. We also deal with pencils whose conic-line curves are in general position.