On the tautological ring of a Jacobian modulo rational equivalence

TL;DR

This paper investigates the structure of the tautological ring of a Jacobian, deriving relations among cycles under specific divisor conditions, with applications to special classes of algebraic curves.

Contribution

It introduces new relations in the tautological ring of Jacobians when the canonical divisor is a multiple of a divisor in a base point free linear system.

Findings

Derived relations between tautological cycles under specific divisor conditions

Applied results to curves with degree d coverings of P^1 with ramification points of order d

Provided insights into the tautological ring structure of hyperelliptic curves

Abstract

We consider the Chow ring with rational coefficients of the Jacobian of a curve. Assume D is a divisor in a base point free g^r_d of the curve such that the canonical divisor K is a multiple of the divisor D. We find relations between tautological cycles. We give applications for curves having a degree d covering of P^1 whose ramification points are all of order d, and then for hyperelliptic curves.