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<p>In this paper, we study the restrictions on the number <i>m</i> 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 <i>m</i> corresponding to the number <i>p</i> of members of concurrent lines; in particular, we recover the universal bound <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(m\le 6\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>≤</mo> <mn>6</mn> </mrow> </math></EquationSource> </InlineEquation> 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 <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(m=6\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>=</mo> <mn>6</mn> </mrow> </math></EquationSource> </InlineEquation>, the conic-line members are in general position and determine their irreducible decomposition.</p>

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On the Number of Conic-Line Curves in a Pencil

  • 期刊论文

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  • 发表日期：

    2026年5月13日

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  • Hasan Suluyer

摘要

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 \(m\le 6\) 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.