Sections of Jacobian fibrations over lines

2025/08/13 by János Kollár, Kollár, János, Giulia Saccà +1
Mathematics · #Algebraic Geometry (math.AG) #FOS: Mathematics #math.AG

paper · pdf · doi:10.48550/arxiv.2508.09819

arxiv created 2026/08/01 · arxiv updated 2026/08/04

Abstract

Let |H| be a linear system on a smooth surface S. We study the cohomology classes of sections of the universal Jacobian over lines in |H|. When S is a K3 surface, the universal compactified Jacobian is a hyperkähler manifold, and we do not see how to reconcile our results with some of the steps in [BKV25]. v.2: Confusing typo fixed: H^g should be H^2g. v3: Note added: Conjecture 7 is now corrected and solved by Pascal Autissier and Andrea Fanelli (On a conjecture by Kollár and Saccà, arXiv:2606.26983).

Citations

• Abelian fiber spaces and their Tate-Shafarevich twists with sections
• Compactifying Lagrangian fibrations
• Shafarevich-Tate groups of holomorphic Lagrangian fibrations II
• Sections of Lagrangian fibrations on holomorphic symplectic manifolds
• Shafarevich-Tate groups of holomorphic Lagrangian fibrations
• Shioda-Tate formula for an abelian fibered variety and applications
• Euler number of the compactified Jacobian and multiplicity of rational curves
• Desingularized moduli spaces of sheaves on a K3, II

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