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arxiv: 2501.08624 · v3 · submitted 2025-01-15 · 🧮 math.AG

A Semi-Orthogonal Decomposition Theorem for Weighted Blowups

Oliver Li This is my paper

Pith reviewed 2026-05-23 05:38 UTC · model grok-4.3

classification 🧮 math.AG
keywords semi-orthogonal decompositionweighted blowupalgebraic stacksKoszul-regular centrederived categoriesOrlov theoremBergh-Schnürer techniques
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The pith

A semi-orthogonal decomposition exists for the weighted blowup of an algebraic stack along a Koszul-regular weighted centre.

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

The paper proves that the derived category of a weighted blowup of an algebraic stack along a Koszul-regular weighted centre decomposes into a semi-orthogonal sequence. This extends Orlov's theorem on ordinary blowups to the weighted and stack-theoretic setting. The result matters because such decompositions let researchers break down derived categories into simpler pieces that can be analyzed inductively. The proof adapts methods from Bergh and Schnürer to accommodate the weights and the stack structure.

Core claim

We establish a semi-orthogonal decomposition for the weighted blowup of an algebraic stack along a Koszul-regular weighted centre, generalising the classic result of Orlov. Our approach is based on the work of Bergh-Schnürer.

What carries the argument

The semi-orthogonal decomposition of the derived category of the weighted blowup, obtained by adapting Bergh-Schnürer techniques to the weighted stacky case.

If this is right

  • The derived category of the weighted blowup decomposes semi-orthogonally into admissible subcategories corresponding to the base stack and weighted shifts of the centre.
  • The result applies to algebraic stacks rather than only to schemes or varieties.
  • The decomposition holds whenever the centre satisfies the Koszul-regularity condition.
  • The proof technique carries over from the unweighted case via direct adaptation of Bergh-Schnürer methods.

Where Pith is reading between the lines

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

  • The theorem supplies a new inductive tool for computing invariants of derived categories on stacks obtained by successive weighted blowups.
  • It may simplify the study of birational maps between stacks by reducing questions about one stack to questions about its weighted blowup and centre.
  • Concrete low-dimensional examples such as weighted projective stacks could be used to verify the length and the explicit form of the semi-orthogonal sequence.

Load-bearing premise

The weighted centre is Koszul-regular and the Bergh-Schnürer techniques extend directly to the weighted stacky setting without additional obstructions.

What would settle it

An explicit computation of the derived category for a concrete weighted blowup of a stack along a Koszul-regular centre that fails to exhibit the predicted semi-orthogonal pieces.

read the original abstract

We establish a semi-orthogonal decomposition for the weighted blowup of an algebraic stack along a Koszul-regular weighted centre, generalising the classic result of Orlov. Our approach is based on the work of Bergh-Schn\"urer.

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 manuscript establishes a semi-orthogonal decomposition for the weighted blowup of an algebraic stack along a Koszul-regular weighted centre, generalising Orlov's classic result via the techniques of Bergh-Schnürer.

Significance. If verified, the result would extend a standard tool from derived categories of schemes to the weighted stacky setting, providing a concrete generalisation that builds directly on prior independent work. No machine-checked proofs or parameter-free derivations are indicated in the available text.

major comments (1)
  1. The provided text consists only of the abstract; no sections, lemmas, or proof steps are supplied, preventing verification of whether the Bergh-Schnürer techniques extend without obstruction to the weighted stacky case as claimed.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report. We address the single major comment below by clarifying the availability of the full manuscript.

read point-by-point responses

  1. Referee: The provided text consists only of the abstract; no sections, lemmas, or proof steps are supplied, preventing verification of whether the Bergh-Schnürer techniques extend without obstruction to the weighted stacky case as claimed.

    Authors: The complete manuscript, including all sections, lemmas, and full proof steps, is available on arXiv:2501.08624. The review materials appear to have included only the abstract excerpt; the full text applies Bergh-Schnürer methods to weighted blowups of algebraic stacks along Koszul-regular centers and verifies that the techniques extend without obstruction, with explicit constructions of the semi-orthogonal decomposition generalizing Orlov's result. revision: no

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper establishes a semi-orthogonal decomposition for weighted blowups of algebraic stacks along Koszul-regular centres by generalising Orlov's result via Bergh-Schnürer techniques. No self-citations, self-definitional steps, fitted inputs presented as predictions, or ansatz smuggling appear in the provided abstract or context. The central claim is framed as an extension of independent prior results without reduction to the paper's own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no information on free parameters, axioms, or invented entities; the Koszul-regularity condition is stated as a hypothesis but its precise formulation is not given.

pith-pipeline@v0.9.0 · 5540 in / 1037 out tokens · 31455 ms · 2026-05-23T05:38:24.736413+00:00 · methodology

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Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages · 1 internal anchor

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