Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">${\mathcal {A}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">A</mi> </math></EquationSource> </InlineEquation> be a free arrangement of d lines in the complex projective plane, with exponents <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$d_1\le d_2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>d</mi> <mn>1</mn> </msub> <mo>≤</mo> <msub> <mi>d</mi> <mn>2</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>. Let m be the maximal multiplicity of points in <InlineEquation ID="IEq3"> <EquationSource Format="TEX">${\mathcal {A}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">A</mi> </math></EquationSource> </InlineEquation>. In this note, we describe first the simple cases <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$d_1 \le m$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>d</mi> <mn>1</mn> </msub> <mo>≤</mo> <mi>m</mi> </mrow> </math></EquationSource> </InlineEquation>. Then, we study the case <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$d_1=m+1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>d</mi> <mn>1</mn> </msub> <mo>=</mo> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and describe which line arrangements can occur by deleting or adding a line to <InlineEquation ID="IEq6"> <EquationSource Format="TEX">${\mathcal {A}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">A</mi> </math></EquationSource> </InlineEquation>. When <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$d \le 14$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>≤</mo> <mn>14</mn> </mrow> </math></EquationSource> </InlineEquation>, there are only two free arrangements with <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$d_1=m+2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>d</mi> <mn>1</mn> </msub> <mo>=</mo> <mi>m</mi> <mo>+</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, namely one with degree 13 and the other with degree 14. We study their geometries in order to deepen our understanding of the structure of free line arrangements in general.

Free line arrangements with low maximal multiplicity

期刊论文
发表日期：

2026年5月4日

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Alexandru Dimca,
Lukas Kühne,
Piotr Pokora

摘要

Let ${\mathcal {A}}$ $A$ be a free arrangement of d lines in the complex projective plane, with exponents $d_1\le d_2$ $d_{1} \leq d_{2}$ . Let m be the maximal multiplicity of points in ${\mathcal {A}}$ $A$ . In this note, we describe first the simple cases $d_1 \le m$ $d_{1} \leq m$ . Then, we study the case $d_1=m+1$ $d_{1} = m + 1$ and describe which line arrangements can occur by deleting or adding a line to ${\mathcal {A}}$ $A$ . When $d \le 14$ $d \leq 14$ , there are only two free arrangements with $d_1=m+2$ $d_{1} = m + 2$ , namely one with degree 13 and the other with degree 14. We study their geometries in order to deepen our understanding of the structure of free line arrangements in general.