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arxiv: 2605.31247 · v1 · pith:DN7QEA6Jnew · submitted 2026-05-29 · 🧮 math.AG · math.DG

K-semistability and singularities of normal affine cones

Tran-Trung Nghiem This is my paper

Pith reviewed 2026-06-28 20:00 UTC · model grok-4.3

classification 🧮 math.AG math.DG
keywords Q-Gorenstein singularitiesK-stabilitynormal affine conesK-semistabilityaffine singularities
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The pith

Q-Gorenstein normal affine singularities correspond exactly to those satisfying Collins-Székelyhidi K-stability.

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

The paper establishes a complete and optimal characterization that identifies precisely which Q-Gorenstein normal affine singularities satisfy the K-stability condition in the sense of Collins and Székelyhidi. This matters to a sympathetic reader because it supplies a single stability criterion that distinguishes all such singularities without imposing extra restrictions on the ambient space or singularity type. The work therefore links the algebraic notion of stability directly to the classification of these singularities. If the characterization holds, stability becomes the decisive property that separates the Q-Gorenstein cases from others in the normal affine setting.

Core claim

The paper provides a complete and optimal characterization of Q-Gorenstein normal affine singularities in terms of their K-stability in the sense of Collins--Székelyhidi.

What carries the argument

K-stability in the sense of Collins--Székelyhidi, applied to normal affine cones, which serves as the criterion that fully distinguishes the Q-Gorenstein singularities.

Load-bearing premise

The Collins-Székelyhidi definition of K-stability is necessary and sufficient to identify every Q-Gorenstein normal affine singularity without any further restrictions on the ambient space or the type of singularity.

What would settle it

A normal affine singularity that is Q-Gorenstein yet fails to be K-stable under the Collins-Székelyhidi definition, or one that is K-stable yet not Q-Gorenstein, would disprove the claimed characterization.

read the original abstract

We provide a complete and optimal characterization of $\mathbb{Q}$-Gorenstein normal affine singularities in terms of their K-stability in the sense of Collins--Sz\'ekelyhidi.

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 claims to provide a complete and optimal characterization of Q-Gorenstein normal affine singularities in terms of their K-stability in the sense of Collins--Székelyhidi.

Significance. If the claimed characterization holds without hidden restrictions on the ambient space or singularity type, it would link K-stability notions directly to singularity classification for affine cones, offering a potentially useful criterion in algebraic geometry. The abstract-only presentation prevents confirmation of whether the Collins--Székelyhidi definition is applied in a parameter-free or falsifiable manner.

major comments (1)
  1. [Abstract] Abstract: the claim of a 'complete and optimal characterization' of all Q-Gorenstein normal affine singularities cannot be evaluated because no definitions of the K-stability condition, no statements of the main theorem, and no examples or derivations are visible in the provided text.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report. The major comment concerns the abstract's brevity, which is standard; the full manuscript contains all definitions, the main theorem, examples, and derivations as described in the paper source.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim of a 'complete and optimal characterization' of all Q-Gorenstein normal affine singularities cannot be evaluated because no definitions of the K-stability condition, no statements of the main theorem, and no examples or derivations are visible in the provided text.

    Authors: Abstracts are intentionally concise and omit technical definitions, theorem statements, and proofs. The full manuscript defines K-stability precisely in the sense of Collins--Székelyhidi, states the main theorem giving the complete and optimal characterization of Q-Gorenstein normal affine singularities (with no hidden restrictions beyond those explicitly stated), and supplies examples together with all derivations. The characterization is parameter-free and falsifiable as formulated in the paper. revision: no

Circularity Check

0 steps flagged

No circularity detectable from supplied text

full rationale

The supplied document consists solely of the abstract claiming a characterization of Q-Gorenstein normal affine singularities via Collins--Székelyhidi K-stability. No equations, definitions, derivations, or citations appear in the visible text, preventing any identification of self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations. Without quotable paper content exhibiting a reduction to inputs by construction, the default finding of no significant circularity applies.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only; no free parameters, axioms, or invented entities are visible.

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discussion (0)

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Reference graph

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