We show that if C is a smooth projective curve and <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2025_2134_Article_IEq1.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="10" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathfrak {d}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="fraktur">d</mi> </math></EquationSource> </InlineEquation> is a <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2025_2134_Article_IEq2.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="25" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathfrak {g}^{n}_{2n}$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mrow> <mi mathvariant="fraktur">g</mi> </mrow> <mrow> <mn>2</mn> <mi>n</mi> </mrow> <mi>n</mi> </msubsup> </math></EquationSource> </InlineEquation> on C, then we obtain a rational map <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2025_2134_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="110" /> </InlineMediaObject> <EquationSource Format="TEX">$\textrm{Sym}^{n}(C)\dashrightarrow \mathfrak {d}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mtext>Sym</mtext> <mi>n</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mrow> <mo>⤏</mo> <mi mathvariant="fraktur">d</mi> </mrow> </math></EquationSource> </InlineEquation> whose fibers can be related in an interesting way to Gunning multisecants of the Kummer variety of JC. This generalizes previous work done by the first author with Codogni and Salvati Manni.

A note on multisecants of the Kummer variety of a Jacobian

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发表日期：

2025年5月15日

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Robert Auffarth,
Sebastian Rahausen

摘要

We show that if C is a smooth projective curve and $\mathfrak {d}$ $d$ is a $\mathfrak {g}^{n}_{2n}$ $g_{2 n}^{n}$ on C, then we obtain a rational map $\textrm{Sym}^{n}(C)\dashrightarrow \mathfrak {d}$ ${Sym}^{n} (C) ⤏ d$ whose fibers can be related in an interesting way to Gunning multisecants of the Kummer variety of JC. This generalizes previous work done by the first author with Codogni and Salvati Manni.