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arxiv: 2405.11860 · v2 · submitted 2024-05-20 · 🧮 math.AG

Fourier-Mukai transforms and normalisation of nodal curves

Emilio Franco , Robert Hanson , Johannes Horn , Andr\'e Oliveira This is my paper

Pith reviewed 2026-05-24 01:37 UTC · model grok-4.3

classification 🧮 math.AG
keywords Poincaré sheafFourier-Mukai transformnodal curvecompactified Jacobianpartial normalisationparabolic modulesmirror symmetry
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The pith

The Poincaré sheaf on the singular locus of a nodal curve's compactified Jacobian restricts to the sheaf on each partial normalisation via parabolic modules.

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

The paper proves that Arinkin's Poincaré sheaf P_C on the singular locus of the compactified Jacobian of an integral nodal curve C can be recovered from the Poincaré sheaf P_Σ on each partial normalisation Σ of C. Strata in the singular locus are indexed by these normalisations, and the restriction of P_C to each stratum equals an expression built from P_Σ. The proof proceeds by passing through the moduli space of parabolic modules, which supplies an intermediate geometry that matches sheaf data on C with data on Σ. This produces an explicit relation between the Fourier-Mukai transforms induced by P_C and by P_Σ. The formulae are presented for later use in mirror symmetry statements on singular loci of Hitchin systems.

Core claim

The central claim is that the Poincaré sheaf P_C restricted to each stratum of Sing(Jac_C) indexed by a partial normalisation Σ → C can be expressed through the Poincaré sheaf P_Σ. The relation between the associated Fourier-Mukai transforms follows directly from this expression. The proof uses the moduli space of parabolic modules of Bhosle and Cook as the geometry that intertwines the sheaf data on the two curves.

What carries the argument

The moduli space of parabolic modules, which supplies an intermediate geometry that intertwines sheaf data on the nodal curve C and its partial normalisation Σ, allowing the restriction of P_C to each stratum to be written in terms of P_Σ.

If this is right

  • The Fourier-Mukai transform defined by P_C is related on each stratum to the transform defined by P_Σ.
  • Properties of the transforms on the singular compactified Jacobian reduce to corresponding properties on the Jacobian of the normalised curve.
  • The resulting formulae can be inserted into the study of mirror symmetry for the singular loci of Hitchin systems.
  • The same intertwining geometry can be used to compare other sheaf-theoretic invariants on C and on Σ.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The relation permits inductive calculation of the transforms by successively normalising the curve one node at a time.
  • The parabolic-module bridge may apply to moduli problems for sheaves on curves with marked points or other parabolic structures.
  • Similar restriction statements could hold for Fourier-Mukai transforms attached to other moduli spaces stratified by partial normalisations.

Load-bearing premise

The singular locus of the compactified Jacobian is stratified by partial normalisations of the curve, and the parabolic modules moduli space supplies the geometry that matches the sheaf data on C with the data on Σ.

What would settle it

A concrete nodal curve C together with one of its partial normalisations Σ where the restriction of P_C to the corresponding stratum fails to equal the explicit expression built from P_Σ.

read the original abstract

We study Arinkin's Poincar\'e sheaf $\mathcal{P}_C$ on the singular locus of $\overline{\mathsf{Jac}}_C$, the compactified Jacobian of rank one torsion-free sheaves on an integral nodal projective curve $C$. Each stratum of the singular locus $\mathsf{Sing}(\overline{\mathsf{Jac}}_C)$ is indexed by a partial normalisation $\Sigma \to C$. We prove that the Poincar\'e sheaf $\mathcal{P}_C$ restricted to each stratum can be expressed through the Poincar\'e sheaf $\mathcal{P}_{\Sigma}$, obtaining a relation between Fourier-Mukai transforms associated to $\mathcal{P}_C$ and $\mathcal{P}_{\Sigma}$. Our approach uses an intermediate geometry: the moduli space of parabolic modules of Bhosle and Cook, to intertwine sheaf data over the two curves. In a sequel, our formulae are used to study mirror symmetry in singular loci of Hitchin systems.

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

0 major / 2 minor

Summary. The manuscript proves that Arinkin's Poincaré sheaf P_C on the singular locus of the compactified Jacobian of an integral nodal curve C, when restricted to each stratum of Sing(Jac_C) indexed by a partial normalisation Σ → C, can be expressed in terms of the Poincaré sheaf P_Σ. The proof uses the moduli space of parabolic modules of Bhosle and Cook as an intermediate geometry to intertwine the sheaf data on C and Σ, thereby obtaining a relation between the Fourier-Mukai transforms associated to P_C and P_Σ. Applications to mirror symmetry in singular loci of Hitchin systems are deferred to a sequel.

Significance. If the stated relation holds, the result supplies an explicit geometric bridge between Fourier-Mukai transforms on a nodal curve and those on its partial normalisations. The intermediate use of Bhosle-Cook parabolic modules provides a concrete intertwining construction that is not circular and follows the standard indexing of strata by partial normalisations; this is a strength for potential computational applications in singular Hitchin systems.

minor comments (2)
  1. [Abstract] Abstract, paragraph on approach: the indexing of strata by partial normalisations Σ → C is asserted to follow from the standard description of the compactified Jacobian, but a one-sentence reminder of the precise correspondence (e.g., via the multiplicity of nodes or the type of torsion-free sheaf) would make the setup self-contained for readers.
  2. [Abstract] Abstract: the notation alternates between overline{sf Jac}_C and sf Jac_C; adopt a single consistent notation throughout the manuscript and abstract.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript and the recommendation of minor revision. No major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via explicit geometric construction

full rationale

The central result expresses the restriction of Arinkin's Poincaré sheaf P_C to strata of Sing(Jac_C) in terms of P_Σ by means of an explicit intermediate moduli space of parabolic modules (Bhosle-Cook). This construction supplies an independent geometric bridge between the two curves; the indexing of strata by partial normalisations is taken from the standard description of the compactified Jacobian, and the relation is built rather than defined into existence or recovered from a fitted parameter. No load-bearing step reduces to a self-citation chain, an ansatz smuggled via prior work, or a renaming of a known empirical pattern. The argument therefore remains non-circular.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard domain assumptions from algebraic geometry concerning the existence and properties of compactified Jacobians, Poincaré sheaves, and parabolic module moduli spaces; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • domain assumption Arinkin's Poincaré sheaf P_C is well-defined on the singular locus of the compactified Jacobian of a nodal curve
    Invoked as the object under study
  • domain assumption The moduli space of parabolic modules of Bhosle and Cook exists and intertwines sheaf data on C and its partial normalizations
    Used as the intermediate geometry

pith-pipeline@v0.9.0 · 5691 in / 1498 out tokens · 34354 ms · 2026-05-24T01:37:13.859278+00:00 · methodology

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Reference graph

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