Let X be a K3 surface, let C be a smooth curve of genus g on X, and let A be a base point free and primitive line bundle <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_842_Article_IEq1.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">$g_d^r$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>g</mi> <mi>d</mi> <mi>r</mi> </msubsup> </math></EquationSource> </InlineEquation> on C with <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_842_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">$d\ge 4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>≥</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_842_Article_IEq3.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="75" /> </InlineMediaObject> <EquationSource Format="TEX">$r\ge \sqrt{d/2}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>r</mi> <mo>≥</mo> <msqrt> <mrow> <mi>d</mi> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </msqrt> </mrow> </math></EquationSource> </InlineEquation>. In this paper, we prove that if <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_842_Article_IEq4.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="155" /> </InlineMediaObject> <EquationSource Format="TEX">$g>2d-3+(r-1)^2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>g</mi> <mo>></mo> <mn>2</mn> <mi>d</mi> <mo>-</mo> <mn>3</mn> <mo>+</mo> <msup> <mrow> <mo stretchy="false">(</mo> <mi>r</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation>, then there exists a line bundle N on X which is adapted to |C| such that |A| is contained in the linear system <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_842_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="75" /> </InlineMediaObject> <EquationSource Format="TEX">$|N\otimes {\mathcal {O}}_C|$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">|</mo> <mi>N</mi> <mo>⊗</mo> </mrow> <msub> <mi mathvariant="script">O</mi> <mi>C</mi> </msub> <mrow> <mo stretchy="false">|</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_842_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="190" /> </InlineMediaObject> <EquationSource Format="TEX">${{\,\textrm{Cliff}\,}}(N\otimes {\mathcal {O}}_C)\le {{\,\textrm{Cliff}\,}}(A)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mspace width="0.166667em" /> <mtext>Cliff</mtext> <mspace width="0.166667em" /> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>N</mi> <mo>⊗</mo> <msub> <mi mathvariant="script">O</mi> <mi>C</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mo>≤</mo> <mrow> <mspace width="0.166667em" /> <mtext>Cliff</mtext> <mspace width="0.166667em" /> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>.

A remark on the conjecture of Donagi and Morrison

期刊论文
发表日期：

2025年7月24日

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Kenta Watanabe

摘要

Let X be a K3 surface, let C be a smooth curve of genus g on X, and let A be a base point free and primitive line bundle $g_d^r$ $g_{d}^{r}$ on C with $d\ge 4$ $d \geq 4$ and $r\ge \sqrt{d/2}$ $r \geq \sqrt{d / 2}$ . In this paper, we prove that if $g>2d-3+(r-1)^2$ $g > 2 d - 3 + {(r - 1)}^{2}$ , then there exists a line bundle N on X which is adapted to |C| such that |A| is contained in the linear system $|N\otimes {\mathcal {O}}_C|$ $| N \otimes O_{C} |$ , and ${{\,\textrm{Cliff}\,}}(N\otimes {\mathcal {O}}_C)\le {{\,\textrm{Cliff}\,}}(A)$ $Cliff (N \otimes O_{C}) \leq Cliff (A)$ .