On the Number of Conic-Line Curves in a Pencil

In this paper, we study the restrictions on the number m of conic-line curves appearing as special members of pencils of plane curves. Using purely algebraic-geometric and combinatorial arguments, we establish explicit upper bounds on m corresponding to the number p of members of concurrent lines; in particular, we recover the universal bound m≤ 6 m ≤ 6 in these pencils. We further construct a one-parameter family of pencils, such that each pencil in the family contains exactly four conic-line curves. Finally, in the extremal case of a pencil of odd-degree plane curves, we prove that for $$m=6$$ m = 6 , the conic-line members are in general position and determine their irreducible decomposition.