A note on double Danielewski surfaces

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.

• The paper rectifies foundational errors in the isomorphism classification of double Danielewski surfaces by establishing precise divisibility and degree conditions.
• It employs refined divisor arguments to accurately describe the automorphism groups, demonstrating diagonal and affine actions on the coordinate ring.
• The results extend classical cancellation counterexamples with rigorous conditions and explicit examples, clarifying stable non-isomorphism among affine varieties.

Rigorous Analysis of Double Danielewski Surfaces

Introduction

This note delivers a comprehensive rectification and clarification of earlier foundational results on double Danielewski surfaces, a class of affine algebraic varieties introduced to construct explicit counterexamples to the Cancellation Problem. The authors critically review the isomorphism classification and automorphism groups of these surfaces, emphasizing errors in previous proofs and offering new, detailed arguments.

Background and Problem Statement

Classical Danielewski surfaces are affine surfaces of the form

$$V_n = \{(x, y, z) \in \mathbb{A}^3_k : x^n y = z^2 - 1\}.$$

These surfaces played a pivotal role in the investigation of the Cancellation Problem since they provide distinct isomorphism types $V_n$ such that $$V_n \times \mathbb{A}^1 \cong V_m \times \mathbb{A}^1$$ for $n \neq m$, but $V_n \not\cong V_m$.

Double Danielewski surfaces are defined as

$$W_{(d,e)} = \{ (x, y, z, t) \in \mathbb{A}^4_k : x^d y = P(x, z),\ x^e t = Q(x, y, z)\}$$

where $P(X, Z)$ is monic in $Z$ and $Q(X, Y, Z)$ is monic in $Y$. While these objects were previously studied, this paper identifies crucial missing hypotheses and gaps in the proofs within the influential work of Gupta and Sen (J. Algebra, 2019), particularly concerning isomorphism classes (Theorem 3.11) and the structure of automorphism groups (Theorem 3.13).

Results and Corrections

Isomorphism Classification

A detailed and corrected version of the isomorphism theorem is provided. The main improvement stems from recognizing and filling in gaps regarding necessary divisibility properties and the behavior of isomorphisms for different degrees of $V_n$0 and $V_n$1. The authors show that for isomorphisms $V_n$2 between double Danielewski algebras as above, with $V_n$3:

• The pairs of degrees $V_n$4 and $V_n$5 match.
• There exist $V_n$6 and $V_n$7 such that $V_n$8 acts linearly (up to translation) on the variables $V_n$9 and $$V_n \times \mathbb{A}^1 \cong V_m \times \mathbb{A}^1$$0.
• The isomorphism type constrains the degrees $$V_n \times \mathbb{A}^1 \cong V_m \times \mathbb{A}^1$$1, and for $$V_n \times \mathbb{A}^1 \cong V_m \times \mathbb{A}^1$$2 one must have $$V_n \times \mathbb{A}^1 \cong V_m \times \mathbb{A}^1$$3, $$V_n \times \mathbb{A}^1 \cong V_m \times \mathbb{A}^1$$4.

Crucially, the corrected proof relies on refined divisor arguments (see Lemmas 2.1 and 2.2), which rule out certain pathological scenarios by tracing the divisibility of specific polynomial combinations through the $$V_n \times \mathbb{A}^1 \cong V_m \times \mathbb{A}^1$$5-adic filtration. These arguments ensure polynomial and monic structure is preserved up to the appropriate degree and provide complete necessary and sufficient conditions for isomorphism.

Automorphism Structure

The authors deduce that automorphisms of these algebras must act diagonally or affinely on the underlying coordinate ring in highly restricted ways, reflecting the tight rigidity imposed by the defining equations. Specifically:

• Automorphisms preserve the coordinate subalgebra $$V_n \times \mathbb{A}^1 \cong V_m \times \mathbb{A}^1$$6 and act by scaling on $$V_n \times \mathbb{A}^1 \cong V_m \times \mathbb{A}^1$$7, with associated invariance properties on ideals defining the surface structure.
• The map induced on $$V_n \times \mathbb{A}^1 \cong V_m \times \mathbb{A}^1$$8 is affine-linear modulo $$V_n \times \mathbb{A}^1 \cong V_m \times \mathbb{A}^1$$9.
• The automorphism group respects all natural gradings and can be described in terms of data dictated entirely by the monic structure of $n \neq m$0 and $n \neq m$1 and the exponents $n \neq m$2.

The results further clarify that, in several edge cases or reductions (e.g., $n \neq m$3 or $n \neq m$4), the surfaces reduce to well-studied (classical) Danielewski or affine cases, and the authors’ classification properly accounts for these reductions.

Explicit Examples and Limiting Cases

Extensive examples and remarks are furnished to illustrate sharpness. For instance:

• The necessity of the $n \neq m$5 hypothesis is illustrated, as in the $n \neq m$6 case, nontrivial transformations altering exponents can exist, invalidating previous claims.
• The cases $n \neq m$7 collapse the variety to $n \neq m$8, consistent with the known structure of affine planes.

Implications and Further Directions

The rigorous revision of the isomorphism and automorphism theorems for double Danielewski surfaces ensures the correctness of the cancellation counterexamples in this family. The arguments extend to the broader study of stably non-isomorphic but non-cancellative affine varieties, and their methods (particularly involving the Makar-Limanov invariant and the interplay between coefficients and monicity) are likely to inform new classes of counterexamples and constructions in affine algebraic geometry.

Further, the formalization and explicit counterexamples provided caution against casual application of isomorphism criteria in the presence of parameterized families, especially when considering inductive or iterative structures involving variable exponents and monic polynomials.

The results suggest several future research directions:

• Investigation of triple or further iterated Danielewski-like structures.
• The development of invariants finer than the Makar-Limanov invariant for more intricate families.
• The analysis of cancellation and stable isomorphism within a larger landscape of non-rational or singular surfaces.

Conclusion

This note supplies essential corrections and strengthened arguments on the structure, isomorphism classes, and automorphism groups of double Danielewski surfaces. The refined lemmas and complete proofs resolve longstanding gaps, restoring rigor to this segment of cancellation theory in affine algebraic geometry. The results also clarify the behavior under specialization to boundary cases, situating double Danielewski surfaces as a robust source of counterexamples and as a testbed for the study of automorphism rigidity and cancellation phenomena.