[Paper Review] Note on curves in a Jacobian

Elisabetta Colombo, Bert van Geemen|ArXiv.org|Feb 17, 1992

Algebraic Geometry and Number Theory1 references3 citations

TL;DR

This paper investigates the image of a curve C under multiplication by n in its Jacobian variety, showing that the subgroup generated by these images in the Chow group modulo algebraic equivalence has rank at most d−1, where d is the gonality of C. The result establishes a deep link between the geometry of curves and the structure of their Jacobians via algebraic cycles.

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.

Motivation & Objective

• To understand the behavior of a curve C under the multiplication-by-n map in its Jacobian variety JC.
• To analyze the image of C in the Chow group modulo algebraic equivalence under this map.
• To determine the rank of the subgroup generated by these images in the Chow group.
• To explore the general action of multiplication-by-n on Chow groups of Jacobians.
• To relate the algebraic cycle structure to geometric invariants like gonality.

Proposed method

• Consider the curve C embedded in its Jacobian JC via the Abel-Jacobi map.
• Apply the multiplication-by-n map [n]: JC → JC to the cycle [C], obtaining n_*[C] in the Chow group.
• Study the subgroup of the Chow group modulo algebraic equivalence generated by the cycles n_*[C] for varying n.
• Use geometric and cohomological techniques to bound the rank of this subgroup.
• Relate the bound to the gonality d of C using linear series and rational maps from C to P^1.
• Leverage known results on algebraic cycles and the structure of Jacobians to derive the rank constraint.

Experimental results

Research questions

• RQ1 What is the rank of the subgroup generated by the images of a curve C under multiplication by n in its Jacobian, in the Chow group modulo algebraic equivalence?
• RQ2 How does the gonality of a curve influence the structure of its algebraic cycles in the Jacobian?
• RQ3 What is the general action of the multiplication-by-n map on the Chow groups of a Jacobian variety?
• RQ4 Can the image of C under [n] be algebraically trivial for some n, and what does this imply about the curve?
• RQ5 What geometric invariants of C control the algebraic independence of the cycles n_*[C]?

Key findings

• The subgroup generated by the cycles n_*[C] in the Chow group modulo algebraic equivalence has rank at most d−1, where d is the gonality of C.
• This bound is sharp in the sense that it is determined by the minimal degree of a non-constant morphism from C to P^1.
• The result holds uniformly across all n ≥ 2, indicating a structural constraint on the cycle classes.
• The analysis reveals that the algebraic cycles n_*[C] are not algebraically independent beyond rank d−1.
• The paper establishes foundational constraints on the action of [n] on Chow groups, suggesting deeper arithmetic and geometric control.