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

TL;DR

This paper explores the relationship between tautological rings of moduli spaces of curves and Jacobians, revealing new relations and properties that connect geometric and motivic aspects, and raises open questions about their fundamental nature.

Contribution

It establishes a link between two types of tautological rings, derives relations from motivic Lefschetz decomposition, and investigates Gorenstein properties, providing new insights and raising open questions.

Findings

Relations between tautological classes derived from motivic Lefschetz decomposition.

Verification of Gorenstein properties for small genera.

Discussion on the potential motivic nature of all tautological relations.

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.