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arxiv: 2604.01842 · v2 · submitted 2026-04-02 · 🧮 math.AG

Genus three Ceresa cycles and limit of archimedean heights

Souvik Goswami , Irene Spelta This is my paper

Pith reviewed 2026-05-13 21:12 UTC · model grok-4.3

classification 🧮 math.AG
keywords Ceresa cyclesgenus three curvesmixed Hodge structuresbiextensionarchimedean heightsDeligne splittingnilpotent orbit
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The pith

For genus three Ceresa cycles the limit of the archimedean height equals the Deligne splitting of the boundary biextension mixed Hodge structure.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies one-parameter variations of biextension mixed Hodge structures that arise geometrically from Ceresa cycles on genus three curves. It applies prior results of Brosnan and Pearlstein on asymptotic heights of nilpotent orbits and shows that, once a parameter is fixed compatibly, the limit height is given exactly by the Deligne splitting of the biextension mixed Hodge structure attached to the cycles at the boundary. This supplies an explicit Hodge-theoretic expression for the height in terms of degeneration data rather than direct archimedean integration. A reader cares because the identification turns a transcendental limit into an algebraic computation that can be carried out in the moduli space of curves.

Core claim

For a one-parameter variation of biextension mixed Hodge structures of geometric origin related to Ceresa cycles associated with curves of genus three, after fixing a parameter, the limit height is given by the Deligne splitting of a biextension mixed Hodge structure associated with cycles in the boundary.

What carries the argument

The Deligne splitting of the biextension mixed Hodge structure associated with cycles in the boundary, which evaluates the limit height of the nilpotent orbit.

Load-bearing premise

The one-parameter variation coming from the Ceresa cycles must be of geometric origin in the precise sense required by the Brosnan-Pearlstein limit formula, and the chosen parameter must be compatible with that origin.

What would settle it

For an explicit one-parameter family of genus three curves, compute the asymptotic height limit directly from the nilpotent orbit and compare it with the value obtained from the Deligne splitting at the boundary; mismatch for a compatible parameter falsifies the equality.

Figures

Figures reproduced from arXiv: 2604.01842 by Irene Spelta, Souvik Goswami.

Figure 1
Figure 1. Figure 1: Biextension diagram In the diagram, we have Q(0) = ker  H 2p |Z| (X; Q(p)) → H2p (X; Q(p)) , and Q(1) = Coker H2p−2 (X; Q(p)) → H2p−2 (|W|; Q(p)) , noticing the relation 2p − 2 = 2(d + 1 − q) − 2 = 2 dim(W). The top horizontal exact sequence in diagram 1 gives the extension class for Z and equals the Abel-Jacobi image of Z, while the left vertical exact sequence gives the dual of the extension class for… view at source ↗
read the original abstract

For a one-parameter variation of biextension mixed Hodge structures, Brosnan and Pearlstein showed that the limit of the asymptotic height of the variation is given by a certain limit height of the nilpotent orbit. This limit height depends on the choice of a parameter. In the case of a variation of geometric origin related to Ceresa cycles associated with curves of genus three, after fixing a parameter, we show that this limit height is given by the Deligne splitting of a biextension mixed Hodge structure associated with cycles in the boundary.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The paper considers one-parameter variations of biextension mixed Hodge structures arising from Ceresa cycles on genus-three curves. Building on Brosnan-Pearlstein, it asserts that after fixing an auxiliary parameter the limit of the archimedean height equals the Deligne splitting of the biextension MHS attached to the boundary cycles.

Significance. If the identification is rigorously justified, the result supplies an explicit geometric computation of a limit height in a case of independent interest in algebraic geometry, linking asymptotic heights to the Deligne splitting of a boundary biextension and thereby furnishing a concrete instance of the general Brosnan-Pearlstein theorem.

major comments (1)
  1. The central claim requires that the Ceresa-cycle variation satisfy the precise geometric-origin hypotheses of Brosnan-Pearlstein (including compatibility of the chosen parameter with the monodromy weight filtration and the boundary biextension). No explicit verification of these conditions appears in the manuscript; without it the identification does not follow from the cited theorem.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for highlighting the need for explicit verification of the Brosnan-Pearlstein hypotheses. We agree that this step is required to make the application of the theorem fully rigorous and will revise the paper accordingly.

read point-by-point responses

  1. Referee: The central claim requires that the Ceresa-cycle variation satisfy the precise geometric-origin hypotheses of Brosnan-Pearlstein (including compatibility of the chosen parameter with the monodromy weight filtration and the boundary biextension). No explicit verification of these conditions appears in the manuscript; without it the identification does not follow from the cited theorem.

    Authors: We agree that an explicit verification of the geometric-origin hypotheses is necessary. In the revised manuscript we will insert a new subsection (placed after the definition of the one-parameter variation) that directly checks: (i) compatibility of the fixed auxiliary parameter with the monodromy weight filtration, and (ii) that the boundary biextension mixed Hodge structure arises in the manner required by Brosnan-Pearlstein. With these verifications added, the identification of the limit height with the Deligne splitting will follow immediately from the cited theorem. revision: yes

Circularity Check

0 steps flagged

No circularity: external theorem applied to specific geometric case

full rationale

The derivation cites the Brosnan-Pearlstein result on limit heights for one-parameter biextension MHS variations as an independent external theorem, then specializes it to the genus-three Ceresa cycle family under the stated geometric-origin assumption after fixing a parameter. The identification of the limit height with the Deligne splitting of the boundary biextension follows from that general theorem rather than reducing to a self-definition, fitted input, or self-citation chain. The parameter choice is part of the setup for the nilpotent orbit and does not force the conclusion by construction. No load-bearing self-citations, ansatzes smuggled via prior work, or renamings of known results appear in the claimed chain.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The claim rests on the Brosnan-Pearlstein theorem for one-parameter variations and on the standard identification of Ceresa cycles with biextension mixed Hodge structures; the parameter choice is an additional free element introduced to make the identification hold.

free parameters (1)
  • auxiliary parameter
    The limit height of the nilpotent orbit depends on this parameter; the paper fixes a value so that the geometric Ceresa case matches the Deligne splitting.
axioms (2)
  • standard math Brosnan-Pearlstein theorem on the limit of asymptotic height for one-parameter variations of biextension mixed Hodge structures
    The paper invokes this result as the starting point for the limit computation.
  • domain assumption Ceresa cycles on genus-three curves give rise to biextension mixed Hodge structures of geometric origin
    This geometric origin is required to apply the limit formula in the stated way.

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