Covers of curves, Ceresa cycles, and Unlikely intersections
Tejasi Bhatnagar, Sheela Devadas, Toren D'Nelly-Warady, Padmavathi Srinivasan

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 of genus over a characteristic zero field. We prove that the Ceresa cycle of a very general ramified cover of 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 -dimensional and -dimensional families of genus 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"…
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Commutative Algebra and Its Applications · Advanced Differential Equations and Dynamical Systems
