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arxiv: 2604.26204 · v1 · submitted 2026-04-29 · 🧮 math.AG

Stability and Fourier-Mukai transforms on an eliptic surface

Kota Yoshioka This is my paper

Pith reviewed 2026-05-07 13:16 UTC · model grok-4.3

classification 🧮 math.AG MSC 14J2714F0518E30
keywords stability conditioncoherent sheafelliptic surfaceFourier-Mukai transformderived categoryalgebraic geometryrelative transformmoduli space
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The pith

A stability condition is introduced for coherent sheaves on an elliptic surface and shown to interact in a controlled way with relative Fourier-Mukai transforms.

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

The paper defines a stability condition specifically adapted to coherent sheaves on an elliptic surface. It then examines how objects satisfying this condition behave when pushed forward or pulled back by the relative Fourier-Mukai transform induced by the elliptic fibration. A reader would care because stability conditions organize moduli problems and wall-crossing in the derived category, so compatibility with a natural equivalence like the relative FM transform would let one move questions about stable sheaves from one side of the fibration to the other. The work therefore aims to make the geometry of sheaves on elliptic surfaces more computable by exploiting the fibration structure.

Core claim

A stability condition is defined for coherent sheaves associated to an elliptic surface; relative Fourier-Mukai transforms then map stable sheaves to stable sheaves in a manner that respects the elliptic fibration, thereby allowing stability questions to be transferred across the equivalence of derived categories.

What carries the argument

The relative Fourier-Mukai transform attached to the elliptic fibration, which furnishes an equivalence of derived categories and is required to preserve the newly defined stability condition.

If this is right

  • Stable sheaves are sent to stable sheaves by the relative Fourier-Mukai transform.
  • Moduli spaces of stable sheaves on the elliptic surface are related by the transform, allowing transfer of geometric properties.
  • Invariants attached to stable sheaves can be computed on one side of the fibration and moved to the other.
  • Wall-crossing phenomena for the stability condition become accessible through the derived equivalence.

Where Pith is reading between the lines

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

  • The construction may supply a concrete model for Bridgeland stability on the derived category of an elliptic surface.
  • The same stability could be used to compare moduli spaces of sheaves before and after the transform, yielding relations among their Euler characteristics or other numerical invariants.
  • If the condition extends to families, it might produce a wall-crossing formula that respects the elliptic fibration.

Load-bearing premise

The introduced stability condition must be well-defined for coherent sheaves on the elliptic surface and must remain compatible with the elliptic fibration and the action of the relative Fourier-Mukai transform.

What would settle it

An explicit coherent sheaf on a concrete elliptic surface that satisfies the stability condition but whose image under the relative Fourier-Mukai transform fails to be stable, or a sheaf for which the stability condition itself cannot be consistently defined.

read the original abstract

We shall introduce a stability condition for a coherent sheaf associated to an elliptic surface. Then we study the behavior under relative Fourier-Mukai transforms.

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 / 1 minor

Summary. The manuscript introduces a stability condition for coherent sheaves on an elliptic surface and investigates the behavior of this condition under relative Fourier-Mukai transforms.

Significance. This work has the potential to be significant in the field of algebraic geometry as it connects stability conditions with Fourier-Mukai transforms on elliptic surfaces, which may lead to new insights into the derived category and moduli spaces. The construction appears to be parameter-free and based on standard properties of coherent sheaves.

minor comments (1)
  1. The abstract phrasing 'stability condition for a coherent sheaf' should be clarified in the introduction to specify whether this is a Bridgeland-type stability condition on the derived category or a different notion such as slope stability with respect to the elliptic fibration.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, the recognition of its potential significance in connecting stability conditions with relative Fourier-Mukai transforms on elliptic surfaces, and the recommendation for minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity; standard definitional construction with independent proofs

full rationale

The paper defines a stability condition for coherent sheaves on an elliptic surface and proves its compatibility with relative Fourier-Mukai transforms. This follows the standard pattern of introducing a new notion via axioms and then deriving its functorial properties, without any self-referential definitions, fitted parameters renamed as predictions, or load-bearing self-citations that reduce the central claim to prior unverified inputs. The derivation chain consists of geometric constructions in the derived category that are externally verifiable against standard properties of elliptic fibrations and Bridgeland stability, making the result self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no explicit free parameters, axioms, or invented entities are identifiable from the provided text.

pith-pipeline@v0.9.0 · 5291 in / 1083 out tokens · 60686 ms · 2026-05-07T13:16:27.853798+00:00 · methodology

discussion (0)

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Reference graph

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