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Covers of curves, Ceresa cycles, and Unlikely intersections

This paper proves that the Ceresa cycle of a very general ramified cover of a curve is nontorsion by reducing the problem to the torsion property of a related point in the Jacobian via the "relative canonical shadow" and applying the relative Manin–Mumford theorem from unlikely intersection theory.

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

Published 2026-03-03

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CC BY 4.0

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

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are an architect designing a massive, intricate city called The Jacobian. This city is built to represent the shape and structure of a specific type of curve (a twisted loop, like a pretzel or a figure-eight). Inside this city, there are special "landmarks" called Ceresa cycles.

For most cities (curves), these landmarks are permanent, solid structures. But for some very specific, rare cities, these landmarks are actually just illusions or ghosts. In mathematical terms, they are "torsion," meaning if you add the landmark to itself enough times, it vanishes into thin air. For a long time, mathematicians knew that for most complex curves, these landmarks are real and permanent. But they didn't know what happened when you built these curves by stacking or covering one curve on top of another (like wrapping a ribbon around a ball).

This paper is a detective story about finding out if these "ghost landmarks" stay ghosts or become real when you use this stacking method.

The Main Characters

The Curve ( $C$ ): The base shape, like a twisted pretzel.
The Cover ( $\tilde{C}$ ): A new shape created by wrapping the base curve with a "blanket" that has holes or twists in it (a ramified cover).
The Ceresa Cycle: A special mathematical object attached to the curve. If it's "torsion," it's a ghost. If it's "nontorsion," it's a solid, permanent building.
The Relative Canonical Shadow: This is the paper's secret weapon. Imagine the Ceresa cycle is a giant statue casting a shadow. The "Relative Canonical Shadow" is a specific, smaller shadow cast by that statue onto a different part of the city (the Jacobian).
- The Rule: If the big statue (Ceresa cycle) is a ghost, its shadow must also be a ghost (a torsion point).
- The Strategy: Instead of trying to prove the big statue is real directly (which is hard), the authors prove the shadow is real. If the shadow is real, the statue must be real too!

The Big Discovery

The authors prove two main things:

1. The "Very General" Rule:
If you pick a random, complex way to wrap a blanket over a curve (a "very general" cover), the resulting Ceresa cycle is always real (nontorsion). It's not a ghost. This confirms a suspicion that these structures are usually solid, even when built in complicated ways.

2. The "Unlikely Intersection" Detective Work:
The authors then looked at specific families of curves (like a family of curves changing slightly as you turn a dial). They asked: "Is there a specific setting on the dial where the Ceresa cycle turns into a ghost?"

To answer this, they used a concept called Unlikely Intersections.

The Analogy: Imagine you have two runners on a circular track. One runner is running at a steady pace, and the other is running at a different steady pace.
The Question: Will they ever meet at the exact same spot at the exact same time?
The Math: In most cases, if the runners are independent, they will never meet at a "special" spot (a torsion point) unless the track is set up in a very specific, rare way.
The Result: The authors showed that for their specific families of curves, the "shadows" (the runners) are so independent that they never meet at the "ghost" spots, except for a tiny, closed-off set of exceptions (which they proved were actually empty for their examples).

The Two Case Studies (The "Proofs")

To make their theory concrete, they built two specific examples:

Example 1: The 1-Parameter Family (The Single Dial):
They looked at a family of curves defined by a single equation with a variable $t$ . They showed that no matter what value you pick for $t$ (except for a few broken numbers), the Ceresa cycle is real. They did this by showing that two different "shadows" of the cycle are like two runners on a track who are mathematically guaranteed never to stop at the same "ghost" station simultaneously.
Example 2: The 2-Parameter Family (The Two Dials):
They looked at a more complex family with two variables. Here, the "shadows" generated a 3-dimensional space of movement. They proved that the "ghost" spots are so rare in this 3D space that they form a tiny, closed-off island that the curves never actually visit.

Why Does This Matter?

In the world of math, finding "nontorsion" cycles (real, permanent structures) is like finding a new continent. They are rare and hard to find.

The "Ghost" Problem: For a long time, we only knew that Ceresa cycles were ghosts for very simple curves (like hyperelliptic ones).
The Breakthrough: This paper shows that for a huge variety of complex curves (specifically those made by covering other curves), the Ceresa cycles are solid, permanent structures.
The "Zariski Closed" Result: They didn't just say "it's usually real." They proved that the set of exceptions (where it might be a ghost) is a tiny, closed-off area. In math, this means the "real" behavior is the rule, and the "ghost" behavior is the extreme outlier.

Summary in a Nutshell

Think of the Ceresa cycle as a lighthouse.

For some curves, the lighthouse is broken (torsion/ghost).
For most curves, it works (nontorsion/real).
The authors asked: "If we build a lighthouse on top of a complex, twisted tower (a cover), does it still work?"
Answer: Yes! They proved that for almost all such towers, the lighthouse shines brightly. They used a clever trick (looking at the shadow) and a powerful law of probability (unlikely intersections) to prove that the light never goes out, except in a few impossible-to-reach corners of the universe.

This work helps mathematicians understand the deep, hidden architecture of shapes and how they behave when stacked on top of each other, confirming that "solid" structures are the norm, not the exception.

1. Problem Statement

The paper addresses the torsion properties of the Ceresa cycle in the Chow group of the Jacobian of algebraic curves.

Context: For a smooth, projective curve $C$ of genus $g \geq 2$ over a field of characteristic zero, the Ceresa cycle is defined as $Cera(C) := [i_e(C)] - [-1]^*[i_e(C)]$ in the Chow group of homologically trivial 1-cycles, $CH^1_{hom}(J)$ .
The Question: While Ceresa (1983) proved that for a very general complex curve of genus $g \geq 3$ , the Ceresa cycle is nontorsion modulo algebraic equivalence, the situation for specific families of curves (particularly those arising as covers) is less understood. Specifically, the authors investigate whether the Ceresa cycle is nontorsion in the Chow group (a stronger condition than algebraic equivalence) for curves that are ramified covers of a base curve.
Motivation: Understanding the locus $M_g^{tors}$ (curves with torsion Ceresa cycles) is crucial for the Beilinson–Bloch conjectures and the study of Griffiths groups. It is known that $M_g^{tors}$ contains the hyperelliptic locus and has at most countably many irreducible components, but explicit examples where the torsion locus is Zariski closed and of positive codimension are rare.

2. Methodology

The authors employ a strategy combining intersection theory of algebraic cycles with unlikely intersection theory (specifically the Relative Manin–Mumford theorem).

A. The "Relative Canonical Shadow"

Instead of analyzing the Ceresa cycle directly, the authors reduce the problem to studying a specific point on the Jacobian of the base curve.

Definition: For a separable cover $\phi: \tilde{C} \to C$ , they define the relative canonical shadow $D_\phi$ as a divisor class in $Pic^0(C)$ .
Key Proposition (from [ELS]): If the relative canonical shadow $D_\phi$ has infinite order in the Jacobian $J(C)$ , then the Ceresa cycle $Cera(\tilde{C})$ is nontorsion in $CH^1(J(\tilde{C}))$ .
Reduction: The problem shifts from the high-dimensional Jacobian of the cover $\tilde{C}$ to the Jacobian of the base curve $C$ , where the shadow is a point constructed from the branch points of the cover.

B. Unlikely Intersection Theory

To prove that the relative canonical shadow is nontorsion for a "very general" choice of branch points, the authors utilize the Relative Manin–Mumford Theorem (Gao–Habegger).

Setup: They consider a family of covers parameterized by a base $Z$ . The branch points define a section $s$ into a product of the base curve, which maps to the Jacobian via a weighted Abel-Jacobi map.
Theorem Application: If the image of this section generates the ambient abelian scheme and the dimension of the section is strictly less than the relative dimension of the abelian scheme, the Relative Manin–Mumford theorem implies that the set of parameters where the section hits a torsion point is a Zariski closed subset of positive codimension.
Dimensional Analysis: The authors verify that for their specific families, the dimension of the parameter space is small enough relative to the dimension of the generated abelian subvariety to apply this theorem.

C. Explicit Families and Specialization

For explicit 1-parameter and 2-parameter families, the authors:

Construct the relative canonical shadow explicitly.
Push forward the shadow via quotient maps to obtain sections of elliptic fibrations (or products of elliptic curves).
Use Masser–Zannier and Stoll's results on unlikely intersections to show that the simultaneous torsion locus of these sections is empty or finite.
Verify linear independence of sections by specializing to specific rational points and using computational tools (Magma).

3. Key Contributions and Results

Theorem 1: Very General Covers

Result: For a fixed base curve $C$ of genus $g \geq 2$ , the Ceresa cycle of a very general ramified cover $\tilde{C}$ (with specified monodromy) is nontorsion in the Chow group.
Significance: This extends Ceresa's result to covers. Crucially, it holds even if the base curve $C$ itself has a torsion Ceresa cycle (e.g., if $C$ is hyperelliptic).

Theorem 2: Infinitely Many Families

Result: There exist infinitely many families of ramified covers over a varying family of curves $Z$ (where $\dim Z < g$ ) such that the locus of curves with torsion Ceresa cycles is Zariski closed and has positive codimension.
Mechanism: This relies on the existence of "generating" points in the Jacobian (via Habegger's bounded height theorem) to ensure the dimension conditions for the Relative Manin–Mumford theorem are met.

Theorem 3: Explicit 1-Parameter Family (Genus 6)

Family: A family of genus 6 curves $C_t$ defined by $y^{12} = (\frac{x+1}{x-1})^3 (\frac{x+t}{x-t})^4$ .
Result: $Cera(C_t)$ is nontorsion for every $t \in \mathbb{C} \setminus \{0, \pm 1\}$ .
Method: The family admits a $D_{12}$ action. The authors push the relative canonical shadow to two sections of an elliptic fibration. They prove these sections are linearly independent and use Stoll's theorem to show the simultaneous torsion locus is empty.
Context: This family was previously studied by Laga and Shnidman, who showed the Ceresa cycle is torsion in the Griffiths group for all $t$ . This paper proves it is nontorsion in the Chow group for all $t$ , highlighting the distinction between these groups.

Theorem 4: Explicit 2-Parameter Family (Genus 6)

Family: Double covers of a 2-parameter family of genus 3 plane quartics with an $S_3$ action.
Result: The Ceresa cycle is nontorsion for a general point in the family.
Method: The Jacobian of the genus 3 curve decomposes into a product of elliptic curves. The relative canonical shadow generates an abelian scheme of dimension at least 3. Since the parameter space has dimension 2, the Relative Manin–Mumford theorem implies the torsion locus is a proper closed subset.

4. Significance

New Examples of Nontorsion Cycles: The paper provides some of the first explicit examples where the Ceresa cycle is known to be generically nontorsion in the Chow group and where the torsion locus is provably Zariski closed.
Distinction from Griffiths Group: Theorem 3 explicitly demonstrates a case where the Ceresa cycle is torsion in the Griffiths group (as shown by Laga–Shnidman) but nontorsion in the Chow group, clarifying the hierarchy of these invariants.
Application of Unlikely Intersections: The work successfully adapts the Relative Manin–Mumford theorem (a deep result in arithmetic geometry) to the study of Ceresa cycles in families, bridging the gap between Hodge theory and arithmetic intersection theory.
Testing Zilber–Pink Conjectures: The families constructed serve as test cases for generalized Zilber–Pink conjectures regarding special subvarieties in the moduli space of curves $M_g$ . The authors suggest their families are "non-special" in the context of the variation of mixed Hodge structures associated with the Ceresa cycle.

5. Conclusion

The paper establishes that for a wide class of ramified covers, the Ceresa cycle is nontorsion. By reducing the problem to the torsion of a "relative canonical shadow" and applying powerful theorems from unlikely intersection theory, the authors prove that the locus of curves with torsion Ceresa cycles is small (Zariski closed, positive codimension) in these families. This advances the understanding of the arithmetic and geometric properties of the Chow group of Jacobians.