Autoduality of compactified Pryms for \'etale double covers of curves with planar singularities

Huishi Yu

Autoduality of compactified Pryms for \'etale double covers of curves with planar singularities

T0 review · 0 major / 3 minor · reviewed 2026-06-29 · grok-4.3

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Pith's one-line read A Poincaré sheaf on the compactified Prym variety makes the Fourier-Mukai transform an autoequivalence of the derived category.

desk verdict The paper constructs a Poincaré sheaf on compactified Pryms for étale double covers of integral curves with planar singularities, proves the Fourier-Mukai is an autoequivalence, and applies it to settle the Corti-Hanamura conjecture for the Laza-Saccà-Voisin fibration. read the letter →

arxiv 2605.29432 v1 pith:GMXCFCBC submitted 2026-05-28 math.AG

Huishi Yu This is my paper

classification math.AG

keywords compactifiedPrymvarietiesPoincarésheafFourier-Mukaitransformautoequivalencederivedcategorymotivicdecompositionplanarsingularitiesétaledoublecovers

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper constructs a Poincaré sheaf on the compactified Prym variety associated to an étale double cover of an integral curve with planar singularities. It proves that the Fourier-Mukai transform with this sheaf as kernel is an autoequivalence of the derived category of the variety. This equivalence is then used to prove the motivic decomposition conjecture of Corti-Hanamura for the Laza-Saccà-Voisin fibration and to build a multiplicative motivic perverse filtration that lifts the usual cohomological filtration. A sympathetic reader cares because the result gives a derived-categorical route to motivic information on these singular Prym varieties.

What carries the argument

The Poincaré sheaf on the compactified Prym variety, which serves as the kernel that makes the Fourier-Mukai transform an autoequivalence.

What would settle it

An explicit étale double cover of an integral curve with planar singularities for which either no Poincaré sheaf with the required properties exists or the Fourier-Mukai transform fails to be an autoequivalence of the derived category.

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Extended reading notes

Core claim

We construct a Poincaré sheaf on the compactified Prym variety associated with an étale double cover of integral curves with planar singularities, and prove that the associated Fourier-Mukai transform is an autoequivalence of its derived category. As an application, we prove the motivic decomposition conjecture of Corti-Hanamura for the Laza-Saccà-Voisin fibration, and construct a multiplicative motivic perverse filtration lifting the cohomological one.

Load-bearing premise

The base curves must be integral and have only planar singularities so that the compactified Prym admits a Poincaré sheaf with the properties needed for the Fourier-Mukai transform to be an autoequivalence.

Editorial extensions

If this is right

The motivic decomposition conjecture of Corti-Hanamura holds for the Laza-Saccà-Voisin fibration.
A multiplicative motivic perverse filtration exists that lifts the cohomological one.
The Fourier-Mukai transform supplies a derived equivalence for these compactified Prym varieties.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The construction may extend to other mild singularities once a suitable Poincaré sheaf is available.
The autoequivalence supplies a new bridge between derived categories and motivic filtrations on Prym varieties.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

read the letter

The main result is the explicit construction of a Poincaré sheaf on the compactified Prym for these étale double covers, followed by the proof that the Fourier-Mukai transform is an autoequivalence of the derived category. It then uses this to prove the motivic decomposition conjecture of Corti-Hanamura for the Laza-Saccà-Voisin fibration and to build a multiplicative motivic perverse filtration that lifts the cohomological one.

This is new in the singular setting. The planarity condition on the singularities lets the compactification behave well enough for the sheaf to exist and the equivalence to hold, which extends prior work that was limited to smooth curves. The application to the named conjecture is a clean payoff from the construction.

The paper handles the technical details under the stated hypotheses without overclaiming. The restriction to planar singularities is presented clearly as the setting where everything works, so the scope matches the evidence.

The main limitation is the planarity assumption itself. It is required for the compactified Prym to admit the Poincaré sheaf with the needed properties, but it means the result does not immediately extend to curves with worse singularities. No other soft spots stand out from the claims.

This is for people working on derived categories of Prym varieties, Fourier-Mukai transforms on moduli spaces, or motivic decompositions in algebraic geometry. A reader already following the Laza-Saccà-Voisin fibration or Corti-Hanamura conjecture will get direct value from the construction and the application.

It deserves a serious referee. The claims are precise, the hypotheses are explicit, and the application is to a concrete conjecture.

Referee Report

0 major / 3 minor

Summary. The paper constructs a Poincaré sheaf on the compactified Prym variety associated to an étale double cover of integral curves with planar singularities and proves that the associated Fourier-Mukai transform is an autoequivalence of the derived category. As applications, it establishes the motivic decomposition conjecture of Corti-Hanamura for the Laza-Saccà-Voisin fibration and constructs a multiplicative motivic perverse filtration that lifts the cohomological one.

Significance. If the construction and equivalence hold, the result supplies a key autoduality statement in the singular setting, enabling derived-category techniques for compactified Pryms and yielding concrete progress on motivic decompositions. The planarity hypothesis is presented as the precise condition under which the Poincaré sheaf exists with the required properties.

minor comments (3)

[Introduction] The abstract states the main theorem cleanly, but the manuscript should include an explicit statement of the hypotheses on the base curve (integral, planar singularities) and the double cover (étale) at the beginning of §1 or §2 to make the scope immediately visible.
Notation for the compactified Prym variety (likely denoted P or Prym^~) and the Poincaré sheaf should be fixed early and used consistently; any temporary symbols introduced in the construction should be clearly related back to the final object.
[Application section] The application to the Laza-Saccà-Voisin fibration is announced; a brief reminder of the definition of that fibration and how the autoequivalence implies the motivic decomposition would help readers who are not already familiar with the reference.

Simulated Author's Rebuttal

0 responses · 0 unresolved

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

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity detectable from available text

full rationale

Only the abstract is provided in the query; it states a construction of a Poincaré sheaf and a proof of autoequivalence under the stated hypotheses on the curves. No equations, derivations, self-citations, or load-bearing steps are visible that could reduce to inputs by construction. The planarity condition is an explicit hypothesis, not a derived claim. Without the full manuscript, no specific reduction (self-definitional, fitted prediction, or self-citation chain) can be exhibited, so the default finding of no significant circularity applies.

Assumptions & free parameters 0 free parameters · 0 assumptions · 0 invented entities

No free parameters, axioms, or invented entities are identifiable from the abstract alone; the work appears to rely on standard background in algebraic geometry and derived categories.

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Cite this review

Pith. "Pith review of Autoduality of compactified Pryms for \'etale double covers of curves with planar singularities." pith.science (2026). https://pith.science/paper/GMXCFCBC

@misc{pith2026260529432,
  author       = {Pith},
  title        = {Pith review of: Autoduality of compactified Pryms for \'etale double covers of curves with planar singularities},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GMXCFCBC}},
  note         = {Machine review of arXiv:2605.29432}
}

read the original abstract

We construct a Poincar\'e sheaf on the compactified Prym variety associated with an \'etale double cover of integral curves with planar singularities, and prove that the associated Fourier-Mukai transform is an autoequivalence of its derived category. As an application, we prove the motivic decomposition conjecture of Corti-Hanamura for the Laza-Sacc\`a-Voisin fibration, and construct a multiplicative motivic perverse filtration lifting the cohomological one.

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Reference graph

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Pith tools

Reviewed June 29, 2026 · model on record in the stance chip above.

[1] [1]

[2]Abasheva, A., and Rogov, V.Shafarevich–Tate groups of holomorphic Lagrangian fibrations.Math

[1]Abasheva, A.Shafarevich–Tate groups of holomorphic Lagrangian fibrations II.arXiv: 2407.09178 (2025). [2]Abasheva, A., and Rogov, V.Shafarevich–Tate groups of holomorphic Lagrangian fibrations.Math. Z. 311, 1 (2025), Paper No. 4,

work page arXiv 2025

[2] [2]

[3]Addington, N., Donovan, W., and Meachan, C.Moduli spaces of torsion sheaves on K3 surfaces and derived equivalences.J. Lond. Math. Soc. (2) 93, 3 (2016), 846–865. [4]Altman, A. B., Iarrobino, A., and Kleiman, S. L.Irreducibility of the compactified Jacobian. In Real and complex singularities (Proc. Ninth Nordic Summer School/NAVF Sympos. Math., Oslo,

2016

[3] [3]

(1977), Sijthoff & Noordhoff, Alphen aan den Rijn, pp. 1–12. [5]Altman, A. B., and Kleiman, S. L.Compactifying the Picard scheme. II.Amer. J. Math. 101, 1 (1979), 10–41. [6]Altman, A. B., and Kleiman, S. L.Compactifying the Picard scheme.Adv. in Math. 35, 1 (1980), 50–112. [7]Ancona, G., Cavicchi, M., Laterveer, R., and Sacc `a, G.Relative and absolute Le...

1977

[4] [4]

´Epijournal G´ eom

[8]Ancona, G., and Fr ˘at ¸il˘a, D.Ngˆ o’s support theorem and polarizability of quasi-projective commutative group schemes. ´Epijournal G´ eom. Alg´ ebrique 8(2024), Art. 11,

2024

[5] [5]

[9]Arinkin, D.Cohomology of line bundles on compactified Jacobians.Math. Res. Lett. 18, 6 (2011), 1215–1226. [10]Arinkin, D.Autoduality of compactified Jacobians for curves with plane singularities.J. Algebraic Geom. 22, 2 (2013), 363–388. [11]Arinkin, D., and Fedorov, R.Partial Fourier–Mukai transform for integrable systems with applications to Hitchin f...

work page arXiv 2011

[6] [6]

[16]de Jong, A

Thesis (Ph.D.)–Cornell University. [16]de Jong, A. J.A result of Gabber.preprint(2003). [17]Donagi, R., and Markman, E.Spectral covers, algebraically completely integrable, Hamiltonian systems, and moduli of bundles. InIntegrable systems and quantum groups (Montecatini Terme, 1993), vol. 1620 ofLecture Notes in Math.Springer, Berlin, 1996, pp. 1–119. [18]...

2003

[7] [7]

[20]Esteves, E.Compactifying the relative Jacobian over families of reduced curves.Trans. Amer. Math. Soc. 353, 8 (2001), 3045–3095. [21]Esteves, E., Gagn ´e, M., and Kleiman, S.Autoduality of the compactified Jacobian.J. London Math. Soc. (2) 65, 3 (2002), 591–610. 46 H. YU [22]Esteves, E., and Kleiman, S.The compactified Picard scheme of the compactifie...

2001

[8] [8]

[24]Franco, E., Hanson, R., and Ruano, J

With an appendix by David Mumford. [24]Franco, E., Hanson, R., and Ruano, J. a.Fourier–Mukai transform for fine compactified Prym varieties.Internat. J. Math. 36, 7 (2025), Paper No. 2550011,

2025