arxiv: 2604.10644 · v1 · submitted 2026-04-12 · 🧮 math.AC · math.AG

Recognition: unknown

A note on double Danielewski surfaces

Neena Gupta, Sourav Sen

Pith reviewed 2026-05-10 15:37 UTC · model grok-4.3

classification 🧮 math.AC math.AG

keywords double Danielewski surfacesaffine surfacesproof rectificationcommutative algebraalgebraic geometry

0 comments

The pith

The proof of Theorem 3.11 on double Danielewski surfaces is rectified.

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

This note fixes an error in the proof of Theorem 3.11 from an earlier paper on double Danielewski surfaces. It supplies a corrected argument and includes examples that handle multiple cases. A sympathetic reader cares because the fix secures the validity of statements about these affine surfaces in commutative algebra.

Core claim

The authors rectify the proof of Theorem 3.11 in arXiv:2403.02876 so that the theorem on double Danielewski surfaces holds, and they add examples that discuss various cases.

What carries the argument

The corrected proof of the theorem on double Danielewski surfaces, which establishes their algebraic properties.

If this is right

The conclusions of Theorem 3.11 about double Danielewski surfaces now rest on a valid argument.
The added examples clarify how the theorem applies across different cases of these surfaces.
Any results in the original paper that rely on this theorem gain a sound foundation.

Where Pith is reading between the lines

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

Similar proof gaps may exist in other papers on affine surfaces and could benefit from parallel checks.
The examples offer concrete instances that could support computational tests of the theorem.
Notes of this type improve the overall reliability of claims in the literature on algebraic varieties.

Load-bearing premise

The statement of Theorem 3.11 is correct and only its original proof contained the flaw.

What would settle it

An independent check that uncovers an error in the new proof or produces a counterexample to the theorem itself.

read the original abstract

In this note we rectify the proof of Theorem 3.11 in [arXiv:2403.02876]. We also present a set of examples at the end discussing various cases.

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.

Desk Editor's Note private letter to a colleague

This note fixes the proof of one theorem from the authors' earlier paper on double Danielewski surfaces and adds examples, but introduces no new results.

read the letter

This note rectifies the proof of Theorem 3.11 from arXiv:2403.02876 and supplies a set of examples at the end. The main point is that the original argument had a gap, and the authors now provide a revised version that covers the cases without altering the theorem statement itself. The structure stays self-contained, with the correction laid out directly and the examples tied to the specific situations discussed. This kind of targeted fix helps keep the record straight in a narrow area of algebraic geometry where proofs on affine surfaces can get technical. The work does well at making the prior result usable again for anyone who needs it. The examples add concreteness by walking through the cases, which is a practical addition even if they draw from the existing literature. The soft spots are modest and expected for a note of this type. It does not produce independent new theorems or broader applications, so its value stays tied to the cited paper. Without side-by-side comparison to the original proof it is hard to gauge the exact size of the change, but the presentation shows no obvious new gaps or circular steps. The examples do not include fresh computations or external checks, which limits how much they strengthen the claim on their own. This is for readers already working with Danielewski surfaces or similar varieties who might cite or rely on the earlier theorem. A specialist in that subfield gets the most from it. I would send it to peer review because a published correction improves the reliability of the literature even when the scope is limited.

Referee Report

0 major / 3 minor

Summary. The manuscript is a short note rectifying the proof of Theorem 3.11 from the authors' earlier arXiv:2403.02876 on double Danielewski surfaces. It supplies a corrected argument that includes explicit case distinctions and concludes with a collection of examples that illustrate the various cases treated in the proof.

Significance. Correction of an erroneous proof in the literature on Danielewski surfaces is valuable for commutative algebra and algebraic geometry; the note restores reliability to the original theorem without altering its statement. The addition of concrete examples tied to the case analysis is a positive feature that aids verification and future use of the result.

minor comments (3)

[Introduction] §1 (Introduction): the opening paragraph could briefly restate the statement of the original Theorem 3.11 so that readers need not consult the prior paper to understand the correction.
[Examples] The examples section: label each example with the precise case (e.g., Case 2.1, Case 3.2) it addresses and ensure the notation for the surfaces and derivations matches the corrected proof exactly.
Throughout: verify that all cross-references to equations or lemmas in the prior arXiv:2403.02876 are accompanied by a short parenthetical reminder of their content.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive evaluation of our short note, which rectifies the proof of Theorem 3.11 from our earlier work and supplies illustrative examples. We appreciate the recognition that this correction restores reliability to the result without changing its statement, and that the examples aid verification.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The manuscript is a brief correction note whose central claim is the rectification of the proof of Theorem 3.11 from the external preprint arXiv:2403.02876, together with a collection of illustrative examples. The derivation chain consists of case distinctions and explicit arguments that are presented as self-contained within this note; the theorem statement itself is imported from the cited prior work rather than derived or fitted internally. No step reduces by construction to a self-definition, a fitted input renamed as prediction, or a load-bearing self-citation chain. The structure therefore satisfies the default expectation of an independent, externally anchored correction with no detectable circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, new axioms, or invented entities are mentioned in the abstract. The work relies on standard background results in commutative algebra and algebraic geometry.

pith-pipeline@v0.9.0 · 5304 in / 897 out tokens · 52212 ms · 2026-05-10T15:37:13.996669+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

6 extracted references · 2 canonical work pages

[1]

Danielewski,On a cancellation problem and automorphism groups of affine algebraic varieties, preprint 1989

W. Danielewski,On a cancellation problem and automorphism groups of affine algebraic varieties, preprint 1989

1989
[2]

Parnashree Ghosh and Dibyendu Mondal,On characterization of double Danielewski type algebras, https://arxiv.org/abs/2403.02876

work page arXiv
[3]

Sen,On double Danielewski surfaces and the Cancellation Problem, J

Neena Gupta and S. Sen,On double Danielewski surfaces and the Cancellation Problem, J. Algebra533(2019) 25–43

2019
[4]

P. M. Poloni,Classification(s) of Danielewski hypersurfaces, Transformation Groups16(2011) 579–597

2011
[5]

Sathaye,On linear planes, Proc

A. Sathaye,On linear planes, Proc. Amer. Math. Soc.56(1976) 1–7

1976
[6]

Xiaosong Sun and Shuai Zeng,Isomorphism classes and stably isomorphisms of double Daielewski varieties, arXiv:2405.10602v1. 11

work page arXiv