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arxiv: 2605.27614 · v2 · pith:H2ADARC5new · submitted 2026-05-26 · 🧮 math.AG

Odd Kn\"orrer periodicity as a double cover

Calum Crossley This is my paper

Pith reviewed 2026-06-29 15:07 UTC · model grok-4.3

classification 🧮 math.AG
keywords derived categoriesmatrix factorizationsbranched double coversKnörrer periodicityquadratic potentialsodd cohomological degreealgebraic geometry
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The pith

The derived category of a branched double cover is equivalent to matrix factorizations of a fiberwise quadratic potential on the associated line bundle when the fiber coordinate has odd degree.

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

The paper proves an equivalence between the derived category of a branched double cover and the category of matrix factorizations for a fiberwise quadratic potential on the line bundle. The equivalence requires that the linear fiber coordinate has odd cohomological degree, which means the matrix factorizations lack the usual even-odd splitting. A sympathetic reader would care because this identifies two different ways to describe the same category, allowing techniques from matrix factorizations to apply directly to double covers.

Core claim

We prove that the derived category of a branched double cover is equivalent to a category of matrix factorizations for a fiberwise quadratic potential on the associated line bundle. This requires the linear fiber coordinate to have odd cohomological degree, so we work with matrix factorizations which no longer have the traditional splitting into even and odd parts.

What carries the argument

The equivalence between the derived category of the branched double cover and matrix factorizations of the quadratic potential, realizing odd Knörrer periodicity as a double cover.

If this is right

  • Properties of matrix factorizations transfer to the derived category of the double cover.
  • The equivalence gives a periodicity result that applies specifically in the odd-degree setting.
  • Derived categories of certain singular varieties can be computed using the matrix factorization side.
  • The construction unifies the description of categories associated to quadratic hypersurfaces.

Where Pith is reading between the lines

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

  • The result may suggest similar equivalences for higher-degree covers or other potentials on line bundles.
  • It connects the double-cover construction to broader questions about how derived categories behave under base change or ramification.
  • Explicit examples could be used to test whether the equivalence preserves additional structures such as Hochschild homology.

Load-bearing premise

The linear fiber coordinate must have odd cohomological degree so that matrix factorizations no longer split into even and odd parts.

What would settle it

Explicit computation of both categories for a concrete example such as the double cover of projective space by a quadratic hypersurface in the total space of a line bundle, checking whether the resulting triangulated categories are equivalent.

Figures

Figures reproduced from arXiv: 2605.27614 by Calum Crossley.

Figure 2.1
Figure 2.1. Figure 2.1: Variations on grading data for Landau–Ginzburg models. Definition 2.4. A Z2-graded Landau–Ginzburg model consists of: (1) A smooth algebraic stack X. (2) A Z2-action on X, called R-charge. (3) A Z2-invariant global function w ∈ Γ(X, OX), called the superpotential. It is evenly-graded if Z2 acts trivially. Definition 2.5. A Z-graded Landau–Ginzburg model consists of: (1) A smooth algebraic stack X. (2) A … view at source ↗
Figure 4.1
Figure 4.1. Figure 4.1: Nodal exoflops [PITH_FULL_IMAGE:figures/full_fig_p021_4_1.png] view at source ↗
read the original abstract

We prove that the derived category of a branched double cover is equivalent to a category of matrix factorizations for a fiberwise quadratic potential on the associated line bundle. This requires the linear fiber coordinate to have odd cohomological degree, so we work with matrix factorizations which no longer have the traditional splitting into even and odd parts.

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 / 0 minor

Summary. The manuscript proves that the derived category of a branched double cover is equivalent to a category of matrix factorizations for a fiberwise quadratic potential on the associated line bundle. This requires the linear fiber coordinate to have odd cohomological degree, so the matrix factorizations lack the traditional even/odd splitting.

Significance. If the equivalence holds, the result extends Knörrer periodicity to an odd-degree setting and supplies a new tool for relating derived categories of double covers to matrix factorizations without parity splitting, which may be useful in algebraic geometry.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for reviewing our manuscript. The provided summary accurately captures the main result concerning the equivalence of the derived category of the branched double cover with the category of matrix factorizations in the odd-degree setting. No specific major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper states a proof of an equivalence between the derived category of a branched double cover and matrix factorizations of a quadratic potential, conditioned on the fiber coordinate having odd cohomological degree. No equations, self-referential definitions, fitted parameters renamed as predictions, or load-bearing self-citations appear in the provided abstract or description. The result is presented as a theorem proved under explicit assumptions using standard tools from algebraic geometry and homological algebra (e.g., extensions of Knörrer periodicity), without reducing to its own inputs by construction. This is the expected outcome for a self-contained proof paper.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No information is available from the abstract to identify free parameters, background axioms, or new postulated entities.

pith-pipeline@v0.9.1-grok · 5558 in / 1084 out tokens · 50004 ms · 2026-06-29T15:07:06.389620+00:00 · methodology

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Reference graph

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