On a combinatorial classification of fine compactified universal Jacobians

Abstract 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 "Equation missing" , 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 $$g\geqslant 2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo>⩾</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> . 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.