Cycles on curves and Jacobians: a tale of two tautological rings

2014/07/08 by Qizheng Yin, Yin, Qizheng
Mathematics · #14C25 #14H10 #14H40 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.1407.2216

openalex publication_date 2014/07/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We connect two notions of tautological ring: one for the moduli space of curves (after Mumford, Faber, etc.), and the other for the Jacobian of a curve (after Beauville, Polishchuk, etc.). The motivic Lefschetz decomposition on the Jacobian side produces relations between tautological classes, leading to results about Faber's Gorenstein conjecture on the curve side. We also relate certain Gorenstein properties on both sides and verify them for small genera. Further, we raise the question whether all tautological relations are motivic, giving a possible explanation why the Gorenstein properties may not hold.

Citations

• Relations in the tautological ring
• Jacobians and symmetric products
• Tautological relations in moduli spaces of weighted pointed curves

Related

• Relations between tautological cycles on Jacobians
• Tautological Cycles on Jacobian Varieties
• Algebraic cycles on the Jacobian of a curve with a g^r_d
• Universal algebraic equivalences between tautological cycles on Jacobians of curves
• The H-tautological ring