What question did this study set out to answer?

The research aims to determine the maximum number of conic-line curves in pencils of plane curves.

May 15, 2026Open Access

On the Number of Conic-Line Curves in a Pencil

Key Points

The research aims to determine the maximum number of conic-line curves in pencils of plane curves.
Applied algebraic-geometric and combinatorial arguments to analyze conic-line curves.
Established explicit upper bounds on the number of curves based on concurrent lines.
Constructed a family of pencils containing exactly four conic-line curves.
Demonstrated that the maximum number of conic-line curves is 6 when the curves are in general position.
Found a specific one-parameter family of pencils with exactly four conic-line curves.
Proved that in extremal cases, conic-line members of odd-degree curves exhibit specific irreducible decompositions.

Abstract

Abstract 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.

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