Genus three Ceresa cycles and limit of archimedean heights

Abstract: For a family of smooth, projective curves of genus three, we interpret the limit of archimedean height pairings of Ceresa cycles associated with the family in terms of the archimedean pairing of algebraic cycles naturally associated with the central fiber of the family.

• The paper establishes that the limit of archimedean height pairings for Ceresa cycles equals the boundary pairing under degeneration.
• It utilizes mixed Hodge theory and biextension structures to compute height jumps and ensure continuity between smooth and singular fibers.
• Results indicate that in both nodal and reducible cases, the vanishing height jump (μ = 0) confirms the functorial behavior of limit mixed Hodge structures.

Degeneration of Genus Three Ceresa Cycles and Limit of Archimedean Heights

Overview

The paper addresses the behavior of archimedean height pairings associated with Ceresa cycles in the context of families of genus three algebraic curves degenerating to singular fibers. The central objective is to interpret the limiting value of archimedean height pairings, as smooth curves degenerate to nodal or reducible curves, within the framework of algebraic cycles supported on boundary components of the compactified Jacobian. By leveraging mixed Hodge theoretic tools, biextension structures, and precise geometric conditions on the family, the authors establish a correspondence between the limiting height and the archimedean height pairing of cycles on the boundary, clarifying ambiguities surrounding height jumps and continuity at the boundary.

Ceresa Cycles and Archimedean Heights

For a smooth projective curve of genus $g \geq 3$, the Ceresa cycle $Z(C) = C - [-1]^*C$ in its Jacobian $JC$ is homologically trivial, yet generically exhibits nontrivial algebraic equivalence. The Abel-Jacobi image of Ceresa cycles and associated height functions provide deep invariants, with archimedean heights encapsulating fundamental arithmetic and Hodge-theoretic data. Specifically, the archimedean height pairing $\Ht(Z,W)$ for two cycles $Z, W \subset JC$ is formulated as an obstruction to $\mathbb{R}$-splitness in specific biextension mixed Hodge structures.

For a family of smooth projective genus three curves $C \to \Delta$ degenerating to a central fiber $C_0$ (either irreducible nodal or reducible), the authors track the Ceresa cycles $Z(C_t)$ and investigate the limit

$\lim_{t \to 0} \Ht(Z(C_t), W(C_t)),$

establishing its interpretation in terms of archimedean height pairings between cycles naturally defined in an appropriate compactification $Z(C) = C - [-1]^*C$0 of the Jacobian of $Z(C) = C - [-1]^*C$1.

Degeneration Scenarios and Mixed Hodge Structures

The analysis focuses on two representative degeneration scenarios for the central fiber $Z(C) = C - [-1]^*C$2:

1. Irreducible Nodal Curve: $Z(C) = C - [-1]^*C$3 contains a single node; for its normalization, the difference of preimages $Z(C) = C - [-1]^*C$4 is torsion in the Jacobian.
2. Reducible Curve: $Z(C) = C - [-1]^*C$5 with two components of genus one and two, joined at a separating node.

For each, the behavior of the limit mixed Hodge structure (MHS) on $Z(C) = C - [-1]^*C$6 can be explicitly described. When the vanishing cycle induces nontrivial monodromy (irreducible nodal case), Deligne and Schmid's theory provide split limit MHS after adjusting the degeneration coordinate, contingent on the torsion condition for $Z(C) = C - [-1]^*C$7. In the reducible case, monodromy is trivial, and the limiting structure is pure.

These MHSs are extended to $Z(C) = C - [-1]^*C$8 using exterior powers, which facilitate the geometric interpretation of heights for Ceresa cycles via biextension structures in relative cohomology. Purity and functoriality ensure that the limiting biextension structures retain crucial compatibility with those arising from smooth fibers.

Main Theorem and Proof Structure

The principal result asserts that, under the stipulated degeneration and base-point choices,

$Z(C) = C - [-1]^*C$9

where $JC$0 are the lifts of the limiting Ceresa cycles to the normalization or appropriate boundary strata of $JC$1, and are homologically trivial.

The proof proceeds as follows:

• Limit Construction: The limit of the biextension variation is characterized, using the functoriality of limit MHSs and the exact sequences arising from relative cohomology with supports.
• Purity and Identification: For the irreducible nodal case, the purity of cohomology with support is established for the singular pair $JC$2, enabling identification of graded pieces in the relative limit MHS.
• Height Continuity and Jump: The archimedean height function, which may admit logarithmic singularities (height jumps), is shown to have a zero height jump in the cases examined, guaranteeing continuity.
• Morphisms of MHS and Height Equality: Morphisms of mixed Hodge structures, strict with respect to filtrations and graded pieces, imply equality of heights between biextension structures on the limit and those on the boundary (using arguments akin to Proposition 2.13).

Strong numerical results include the explicit vanishing of the height jump $JC$3 in this family context. For both degeneration types, the limiting height is exactly computable as the archimedean height pairing of cycles defined on the compactified (or normalized) Jacobian.

Contradictory Claims and Implications

A nuanced insight arises regarding higher regulators and height pairings: the limiting heights for Ceresa cycles do not correspond to higher heights of Collino cycles, contradicting expectations based on normal function limits for regulators of higher cycles. This distinction underscores subtle differences in the behavior of limit cycles and their arithmetic invariants within degenerating families.

These results have implications for the approach to boundary behavior in moduli spaces of curves, particularly for extending arithmetic invariants from $JC$4 to its Deligne-Mumford compactification. The clarification of height jump phenomena and continuity of height functions at the boundary, together with explicit identification of limiting heights, strengthens the foundational understanding of height pairings in geometric and arithmetic settings.

Speculation on Future Developments

The findings open avenues for generalizing the height limit results to higher genus or more intricate degenerations, possibly extending to more general higher cycles and their height pairings. Further developments could refine the functorial construction of limiting biextension structures, its interaction with the tautological aspects of the moduli space, and potential applications in arithmetic intersection theory.

Links between limiting height phenomena, boundary components of moduli spaces, and the structure of mixed Hodge modules could be explored. Moreover, the precise role of torsion conditions, height jumps, and the relation to abstract biextension variations merit deeper examination, with possible extensions to the study of automorphic heights and arithmetic Picard groups.

Conclusion

The paper delivers a comprehensive analysis of the degeneration of archimedean height pairings for Ceresa cycles in genus three. By leveraging precise mixed Hodge theoretic constructions and geometric considerations, it establishes the equality of limiting heights and boundary heights under mild degeneration, clarifies the behavior of height jumps, and highlights subtle distinctions in the limit of arithmetic-geometric invariants on degenerating families of curves. The theoretical framework and results provide a solid foundation for further study of height pairings and degenerations in higher-dimensional algebraic-geometric contexts.