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<p>Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="233_2025_10558_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">\({{\mathbb {K}}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">K</mi> </math></EquationSource> </InlineEquation> be an algebraically closed field of characteristic zero. Many of the geometric properties of the plane curves in <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="233_2025_10558_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">\({{\mathbb {K}}}^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">K</mi> </mrow> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> with one place at infinity can be obtained from the arithmetic of numerical semigroups associated with these curves. In this survey, we recall the main properties of these semigroups as well as the geometric results obtained from these properties. We also address some problems.</p>

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On curves with one place at infinity

  • 期刊论文

  • 发表日期：

    2025年7月29日

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  • Abdallah Assi,
  • Wael Mahboub

摘要

Let \({{\mathbb {K}}}\) K be an algebraically closed field of characteristic zero. Many of the geometric properties of the plane curves in \({{\mathbb {K}}}^2\) K 2 with one place at infinity can be obtained from the arithmetic of numerical semigroups associated with these curves. In this survey, we recall the main properties of these semigroups as well as the geometric results obtained from these properties. We also address some problems.