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Covers of curves, Ceresa cycles, and Unlikely intersections

Tejasi Bhatnagar, Sheela Devadas, Toren D'Nelly-Warady, Padmavathi Srinivasan

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.

Covers of curves, Ceresa cycles, and Unlikely intersections

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" 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.
Paper Structure (8 sections, 21 theorems, 38 equations)

This paper contains 8 sections, 21 theorems, 38 equations.

Table of Contents

  1. Relative canonical shadows of Ceresa cycles for separable covers
  2. Unlikely intersections in abelian varieties
  3. Generating subvarieties of abelian varieties
  4. A very general cover of a hyperbolic curve has nontorsion Ceresa cycle
  5. Infinitely many families with small Ceresa-torsion locus
  6. Explicit families of curves with small Ceresa torsion locus
  7. A $1$-parameter $D_{12}$-family.
  8. Covers of a $2$-parameter $S_3$ family.

Key Result

Theorem 1

Assume $g \geq 2$ and that $\rho$ corresponds to a ramified cover of $C$. Then $\operatorname{Cer}(\widetilde{C})$ is nontorsion in the Chow group for a very general point $[\widetilde{C} \rightarrow C]$ in $\mathcal{H}_{d,r}(C,\rho)({\mathbb C})$.

Theorems & Definitions (47)

  • Theorem 1
  • Remark
  • Remark
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Definition 1.1
  • Remark 1.2
  • Proposition 1.3
  • Remark 1.4
  • ...and 37 more