Tautological classes with twisted coefficients

TL;DR

This paper introduces a new framework for studying tautological classes with twisted coefficients on moduli spaces of curves, providing explicit calculations for low genus cases and applications to the Faber conjecture.

Contribution

It defines Chow groups with twisted coefficients, studies their tautological subgroups, and explicitly computes these structures for genus up to 4, advancing understanding of tautological rings.

Findings

Explicit description of tautological rings for genus ≤ 4

Structural results on twisted commutative algebra of tautological classes

Applications to the Faber conjecture

Abstract

Let $M_g$ be the moduli space of smooth genus $g$ curves. We define a notion of Chow groups of $M_g$ with coefficients in a representation of $Sp(2g)$, and we define a subgroup of tautological classes in these Chow groups with twisted coefficients. Studying the tautological groups of $M_g$ with twisted coefficients is equivalent to studying the tautological rings of all fibered powers $C_g^n$ of the universal curve $C_g \to M_g$ simultaneously. By taking the direct sum over all irreducible representations of the symplectic group in fixed genus, one obtains the structure of a twisted commutative algebra on the tautological classes. We obtain some structural results for this twisted commutative algebra, and we are able to calculate it explicitly when $g \leq 4$. Thus we completely determine the tautological rings of all fibered powers of the universal curve over $M_g$ in these genera. We also give some applications to the Faber conjecture.