Covers of curves, Ceresa cycles, and Unlikely intersections

TL;DR

This paper proves that for a very general ramified cover of a curve, the Ceresa cycle is non-torsion in the Jacobian, using unlikely intersection theory to analyze torsion conditions and providing explicit examples of such curves.

Contribution

It establishes the non-torsion nature of Ceresa cycles for very general ramified covers and introduces a new approach using the relative canonical shadow and unlikely intersection theory.

Findings

Ceresa cycle of a very general ramified cover is non-torsion.

Existence of infinite families of curves with non-torsion Ceresa cycles.

Explicit examples of genus 6 curves with positive codimension torsion locus.

Abstract

Fix a smooth, projective, geometrically integral curve $C$ of genus $g \geq 2$ over a characteristic zero field. We prove that the Ceresa cycle $\mathrm{Cer}(\widetilde{C})$ of a very general ramified cover $\widetilde{C}$ of $C$ is nontorsion in the Chow group of its Jacobian. We also show that there exist infinitely many families of ramified covers of a varying family of curves where a general point of these families corresponds to a curve with nontorsion Ceresa cycle. To illustrate this, we write down two explicit $1$-dimensional and $2$-dimensional families of genus $6$ curves where the locus of curves with torsion Ceresa cycle is Zariski closed and has positive codimension. Our strategy is to reduce the question of whether the Ceresa cycle is torsion to the question of whether a related point on the Jacobian of the curve is torsion. For this, we use the ``relative canonical shadow" of the Ceresa cycle, which is a point in the Jacobian of the curve obtained by intersecting the Ceresa cycle with a natural correspondence arising from the covering map. We combine this with ideas from unlikely intersection theory (namely the relative Manin--Mumford theorem) to study the locus where the relative canonical shadows of the Ceresa cycle become torsion.