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<p>In this note we study complex projective plane curves whose graded module of Jacobian syzygies is generated by its minimal degree component. Examples of such curves include the smooth curves as well as the maximal Tjurina curves. However, this class of curves seems to be surprisingly large. In particular, any line arrangement <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1203_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">\({{\mathcal {A}}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">A</mi> </math></EquationSource> </InlineEquation> of <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1203_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="108" /> </InlineMediaObject> <EquationSource Format="TEX">\(d=2k+1\ge 5 \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>=</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>≥</mo> <mn>5</mn> </mrow> </math></EquationSource> </InlineEquation> lines having only double and triple points is in this class if the number of triple points is <i>k</i> and if they are all situated on a line <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1203_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="51" /> </InlineMediaObject> <EquationSource Format="TEX">\(L \in {{\mathcal {A}}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>L</mi> <mo>∈</mo> <mi mathvariant="script">A</mi> </mrow> </math></EquationSource> </InlineEquation>, see Theorem <InternalRef RefID="FPar31">6.2</InternalRef>.</p>

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Curves with Jacobian syzygies of the same degree

  • 期刊论文

  • 发表日期：

    2025年2月24日

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  • Alexandru Dimca,
  • Gabriel Sticlaru

摘要

In this note we study complex projective plane curves whose graded module of Jacobian syzygies is generated by its minimal degree component. Examples of such curves include the smooth curves as well as the maximal Tjurina curves. However, this class of curves seems to be surprisingly large. In particular, any line arrangement \({{\mathcal {A}}}\) A of \(d=2k+1\ge 5 \) d = 2 k + 1 ≥ 5 lines having only double and triple points is in this class if the number of triple points is k and if they are all situated on a line \(L \in {{\mathcal {A}}}\) L ∈ A , see Theorem 6.2.