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<p>Extending the definition of <i>V</i>-stability conditions, given by Viviani in the recent preprint (On the classification of fine compactified Jacobians of nodal curves, 2023), we introduce the notion of <i>universal stability conditions</i>. Building on results by Pagani and Tommasi (Forum Math Sigma 12, Art. No. e87, 2024), we show that fine compactified universal Jacobians, that is, fine compactified Jacobians over the moduli spaces of stable pointed curves <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="MediaObjects/40879_2025_855_IEq1_HTML.gif" Format="GIF" Height="25" Rendition="HTML" Resolution="120" Type="Linedraw" Width="37" /> </InlineMediaObject> </InlineEquation>, are combinatorially classified by universal stability conditions. We use these stability conditions to show the following. The inclusion of fine compactified universal Jacobians of type (<i>g</i>,&#xa0;<i>n</i>) whose fibres over geometric points are classical, that is, they are constructed by some numerical polarisation, into the class of all fine compactified universal Jacobians, is strict, in general, for any <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(g\geqslant 2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>g</mi> <mo>⩾</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. In particular, given any fixed <i>g</i>, we find the minimal number of marked points <i>n</i> necessary for the existence of such a Jacobian, and we provide explicit examples. This answers a question of Pagani and Tommasi.</p>

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On a combinatorial classification of fine compactified universal Jacobians

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  • 发表日期：

    2025年9月18日

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  • Marco Fava

摘要

Extending the definition of V-stability conditions, given by Viviani in the recent preprint (On the classification of fine compactified Jacobians of nodal curves, 2023), we introduce the notion of universal stability conditions. Building on results by Pagani and Tommasi (Forum Math Sigma 12, Art. No. e87, 2024), we show that fine compactified universal Jacobians, that is, fine compactified Jacobians over the moduli spaces of stable pointed curves , are combinatorially classified by universal stability conditions. We use these stability conditions to show the following. The inclusion of fine compactified universal Jacobians of type (g, n) whose fibres over geometric points are classical, that is, they are constructed by some numerical polarisation, into the class of all fine compactified universal Jacobians, is strict, in general, for any \(g\geqslant 2\) g ⩾ 2 . In particular, given any fixed g, we find the minimal number of marked points n necessary for the existence of such a Jacobian, and we provide explicit examples. This answers a question of Pagani and Tommasi.