Note on curves in a Jacobian

TL;DR

This paper investigates the algebraic cycles generated by a curve in its Jacobian under multiplication, establishing a bound on their rank related to the curve's gonality and exploring the action of multiplication on Chow groups.

Contribution

It provides a new upper bound on the rank of cycles generated by a curve in its Jacobian under multiplication, linked to the gonality of the curve.

Findings

The subgroup generated by n_*C has rank at most d-1.

The paper discusses properties of the n_* action on Chow groups.

Provides insights into the structure of algebraic cycles in Jacobians.

Abstract

For a curve C, viewed as a cycle in its Jacobian, we study its image n_*C under multiplication by n on JC. We prove that the subgroup generated by these cycles, in the Chow group modulo algebraic equivalence, has rank at most d-1, where d is the gonality of C. We also discuss some general facts on the action of n_* on the Chow groups.