On Generalised Danielewski Surfaces over fields of arbitrary characteristic

Debojyoti Saha

REVIEW 3 major objections 5 minor 11 references

On Generalised Danielewski Surfaces over fields of arbitrary characteristic

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read These surfaces have the property that every nontrivial exponential map has invariant ring exactly $K[x]$, forcing the Makar-Limanov and Derksen invariants to coincide with $K[x]$ and yielding new counterexamples to the cancellation problem.

desk verdict A useful extension of Danielewski surfaces to arbitrary fields, but the central invariant proof has a grading gap that needs fixing. read the letter →

arxiv 2506.01457 v1 pith:BD3G6Z3Q submitted 2025-06-02 math.AC

Debojyoti Saha This is my paper

classification math.AC MSC 14R1013A5014R0513A0213B25

keywords CancellationProblemExponentialMapMakar-LimanovinvariantDerksenDanielewskisurfacesstableisomorphismlocallynilpotentderivationsarbitrarycharacteristic

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper studies affine surfaces of the form $A = K[X,Y,Z]/(f(X)Y - \varphi(X,Z))$, where $f$ is monic of degree $r\ge2$ and $\varphi$ is monic in $Z$ of degree $d\ge2$, over a field $K$ of arbitrary characteristic. It argues that every nontrivial exponential map on $A$ — that is, every algebraic $\mathbb{G}_a$-action — has exactly $K[x]$ as its ring of invariants, where $x$ is the image of $X$. If true, this identifies both the Makar-Limanov invariant and the Derksen invariant of every such surface with $K[x]$. The paper then uses that invariant computation to classify isomorphisms between these surfaces completely, and it identifies an infinite subfamily that is pairwise non-isomorphic yet becomes isomorphic after adjoining one variable, giving new counterexamples to the cancellation problem over arbitrary fields.

What carries the argument

The load-bearing object is the exponential map, the ring-theoretic avatar of a $\mathbb{G}_a$-action: a $K$-algebra homomorphism $\phi:A\to A[U]$ satisfying $\phi_V\circ\phi_U=\phi_{U+V}$ and evaluation at $U=0$ equal to the identity. Around this, the proof uses three mechanisms. First, an admissible $\mathbb{Z}$-filtration of $A$, induced by the grading of $D=K[x,f(x)^{-1},z]$, reduces the surface to the graded model $B=K[U,V,W]/(f(U)V-W^d)$ with weights $\mathrm{wt}(U)=0$, $\mathrm{wt}(V)=d$, $\mathrm{wt}(W)=1$. Second, the homogenization theorem for exponential maps carries an invariant of $\phi$ on $A$ to an invariant of a nontrivial homogeneous exponential map on $B$. Third, Proposition 3.3 shows on that graded model that no such homogeneous exponential map can fix $v$, forcing the original invariant to lie in $K[x]$. The isomorphism classification then uses the same invariant ring together with ideal-theoretic manipulations, and the stable-isomorphism family is built by extending an exponential map on $A$ to $A^{[1]}$ and constructing an element $w$ so that $A^{[1]}$ becomes a polynomial ring over the subring that is exactly the other surface.

What would settle it

Find a surface $K[X,Y,Z]/(f(X)Y-\varphi(X,Z))$ with $\deg f\ge2$ and $\deg_Z\varphi\ge2$ carrying a nontrivial exponential map whose invariant ring contains an element outside $K[x]$. Theorem 3.5 predicts no such map exists, so one explicit map of that kind, or a homogeneous exponential map on $K[U,V,W]/(f(U)V-W^d)$ fixing $v$, would settle the central claim negatively.

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Extended reading notes

Core claim

The central claim is Theorem 3.5: under the stated hypotheses, for every nontrivial exponential map $\phi$ on $A$, the invariant ring is $A^\phi = K[x]$. Corollary 3.6 then identifies both $\mathrm{ML}(A)$ and $\mathrm{DK}(A)$ with $K[x]$. The same control over invariants drives the isomorphism classification in Theorem 4.1, which says any isomorphism $T:A_1\to A_2$ sends $x_1$ to $\lambda x_2+\mu$, sends $z_1$ to $\gamma z_2+\delta$, preserves the prime-factor structure and multiplicities of $f$ and $g$, and forces $d_1=d_2$. It also drives Theorem 5.2: for $g$ with no double root and $(\varphi,\varphi_Z)=K[X,Z]$, the surfaces $A_n=K[X,Y,Z]/(X^n g(X)Y-\varphi(X,Z))$ with $n\ge2$ are pairwise non-isomorphic but satisfy $A_n^{[1]}\cong A_m^{[1]}$ for all $n,m$. The paper's final conclusion is that this subfamily supplies pairwise non-isomorphic stably isomorphic surfaces, so cancellation fails inside this class over every field.

Load-bearing premise

The argument depends on the homogenization theorem applying to the chosen filtration, in particular on a degree assignment for the homogenizing variable under which an element of degree zero such as $x$ cannot acquire a nonzero $T$-term; if that convention cannot be fixed, the proof that every invariant ring equals $K[x]$ is not established.

Editorial extensions

If this is right

Every nontrivial $\mathbb{G}_a$-action on these surfaces has the same invariant ring $K[x]$, so the $x$-coordinate is the unique common invariant and the surfaces are ML-rigid in this sense.
Two surfaces in the family are isomorphic only when the polynomials $f$ and $g$ have the same prime factors with the same multiplicities and the same degree $d$ in $Z$, giving a complete isomorphism test in terms of the defining data.
The family $\Sigma=\{A_n:n\ge2\}$ yields infinitely many pairwise non-isomorphic surfaces whose one-variable stabilizations are all isomorphic, so cancellation fails for this class over every field.
For a surface with $f=X^n g(X)$ and $(\varphi,\varphi_Z)=K[X,Z]$, each $A_n$ is stably isomorphic to every other member of the family but never isomorphic to any other member.
Because the invariants are known in full, checking whether two such surfaces are isomorphic reduces to comparing the prime factors of $f$ and $g$, their multiplicities, and a congruence involving $\varphi$, rather than searching for an explicit isomorphism.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If Theorem 3.5 is right, the same filtration-homogenization route should give a complete description of the automorphism group of each surface, since Theorem 4.1 fixes the images of $x$ and $z$ and leaves only the possible $\theta$ terms as freedom.
The stable-isomorphism construction is engineered around the coprimality condition $(\varphi,\varphi_Z)=K[X,Z]$; removing that condition is the natural stress test, and one would predict the family to stop being stably isomorphic when $\varphi$ and $\varphi_Z$ share a fibre above a double root of $f$.
The same mechanism may transfer to higher-dimensional hypersurfaces of the form $f_1\cdots f_s\,Y=\varphi(X,Z)$ or to multi-cylinders, where the graded model carries several weights and the first thing to check would be an analogue of Proposition 3.3.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper does something genuinely new: it pushes the Makar-Limanov and Derksen invariant computation for surfaces f(X)Y = phi(X,Z) from algebraically closed characteristic zero to arbitrary fields and characteristics, and it adds a complete isomorphism classification plus a new stable-isomorphism family giving cancellation counterexamples. The overall architecture is credible and the writing is clear. If the main theorem holds, this is a solid contribution to affine algebraic geometry.

The soft spot is real and load-bearing. In Proposition 3.3, the proof considers a homogeneous exponential map on B = K[U,V,W]/(f(U)V - W^d) with weights wt(u)=0, wt(v)=d, wt(w)=1, but never specifies the degree of the homogenizing variable T in B[T]. The homogeneous-map argument then compares leading T-degree terms and concludes that the leading coefficients must be monomials in v. That conclusion depends on the degree of T. If T has positive degree, the degree constraints force u to be invariant, contradicting nontriviality. If T has degree zero, the leading coefficients are forced to be constants, not monomials in v, so the displayed comparison of leading terms does not follow. The proof needs to fix the degree of T or handle the positive-degree case as an immediate contradiction. Without this, Theorem 3.5, Corollary 3.6, and the isomorphism results that rely on them are not yet established.

I disagree with the reader about Lemma 3.2(i). The inference there is fine: K[x] is algebraically closed in A, A^phi has the same transcendence degree as K[x], so equality follows. The other minor terseness in Lemma 3.2(ii) and Section 5 does not bother me much.

This is a paper for specialists in Ga-actions and cancellation problems. The results are likely correct and important, but the central proof needs a fix. I would send it to a serious referee rather than desk reject, with a clear request to address the grading issue. It is the kind of paper worth discussing in a reading group even while the gap is unresolved.

Referee Report

3 major / 5 minor

Summary. The paper studies exponential maps (G_a-actions) on affine surfaces of the form A = K[X,Y,Z]/(f(X)Y - φ(X,Z)) over an arbitrary field K, with f monic of degree r ≥ 2 and φ monic in Z of degree d ≥ 2. The main claims are: (i) every nontrivial exponential map on A has invariant ring exactly K[x], where x is the image of X (Theorem 3.5), hence ML(A) = DK(A) = K[x] (Corollary 3.6); (ii) a complete classification of isomorphisms between two such surfaces (Theorems 4.1 and 4.2); and (iii) a subfamily Σ = {A_n}_{n≥2} that is pairwise non-isomorphic but stably isomorphic, providing new counterexamples to the cancellation problem (Theorems 5.1 and 5.2). An appendix relates the algebraic surfaces to a geometric definition of Danielewski surfaces. The proofs are largely self-contained and build on standard tools: the Abhyankar-Moh theorem, exponential-map lemmas, and Derksen–Hadas–Makar-Limanov homogenization.

Significance. If the results hold, this paper is a valuable extension of Danielewski-surface theory to arbitrary characteristic and non-algebraically closed fields. The explicit determination of ML and DK invariants, the isomorphism classification, and the new cancellation counterexamples are all significant contributions. The constructions are concrete: the exponential map in Lemma 3.2 and the stable isomorphism in Theorem 5.1 are written out and verifiable. The main caveat is that the proof of the central invariant theorem (Theorem 3.5) relies on a homogeneity argument whose grading convention is unspecified and currently not justified; this affects all subsequent results that use Corollary 3.6. The overall strategy is plausible and the gap appears repairable, but it is load-bearing and must be fixed before the paper is fully convincing.

major comments (3)

[§3, Proposition 3.3] The homogeneity comparison 'grdeg(u) = grdeg(g(v)T^n)' and 'grdeg(w) = grdeg(h(v)T^m)' is asserted without specifying the degree of T in B[T]. If deg T = s, these equations read d·α + n·s = 0 and d·β + m·s = 1, where α = deg_v g and β = deg_v h. For s ≥ 0 these equations are incompatible with n, m > 0 and α, β ≥ 0 (for s = 1 the first has no solution; for s = 0 the second has no solution because d ≥ 2). The argument only has a chance for s < 0, but this is never stated, and the integrality of α and β would need to be checked. Since Proposition 3.3 is the key tool in Theorem 3.5, the grading convention must be fixed and the proof completed under that convention.
[§3, Theorem 3.5] The step 'By Theorem 2.8, \bar{g} = \hat{g} ∈ B^φ' requires that the induced homogeneous exponential map on B satisfies the hypotheses of Proposition 3.3, including the homogeneity condition with respect to the grading on B[T]. As noted in the previous comment, Theorem 2.8 is cited without stating the degree it assigns to the homogenizing variable T, and Proposition 3.3's computation is not valid for all possible degrees. Consequently the proof of Theorem 3.5 and its corollaries is incomplete. Please either state the grading produced by Theorem 2.8 explicitly or give a direct argument showing that the induced map satisfies the needed homogeneity.
[§3, Proposition 3.3] The assertion 'neither u nor w can be in B^φ' is not justified. Inertness alone does not rule out the possibility that u ∈ B^φ while the map is nontrivial. A proper proof would show that if u ∈ B^φ, then w^d = f(u)v ∈ B^φ and hence w ∈ B^φ by inertness, forcing B^φ = B and contradicting nontriviality; the same reasoning applies to w. Please include this argument or an equivalent one.

minor comments (5)

[§3, Theorem 3.5] In the statement of Theorem 3.5, 'k[x]' should be 'K[x]' for consistency with the rest of the paper.
[§5, Theorem 5.1] After the display defining w = (v − s a(x,θ))/x, the text says 'Then by (11), w ∈ R', but the relevant equation is the preceding display, not (11). The equation numbering in this paragraph is off by one.
[§2, Definition 2.7] The definition of a homogeneous exponential map says that U becomes a homogeneous element but does not specify its degree. Since the degree of the analogous variable T is exactly what is left unspecified in Proposition 3.3, please state the convention here.
[§3, Lemma 3.4] The proof that the filtration {A_i} is admissible is compressed. A few more details on how the normal form in Lemma 3.1(iv) yields the required monomial decomposition for each A_i would help the reader.
[§3, Lemma 3.2(i) and Lemma 3.1(ii)] Lemma 3.1(ii) is cited from [3] with the assertion that the proof is characteristic-independent, but the argument is not reproduced. Since this lemma is used to prove Lemma 3.2(i), please include a self-contained proof or provide a precise reference to a statement covering arbitrary characteristic.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the main theorems are derived from external standard results and internal constructions, with no fitted parameters, definitional identifications, or self-citation chains used as load-bearing support.

full rationale

The paper's central claim, Theorem 3.5, is proved by applying the homogenization theorem (Theorem 2.8, attributed to Derksen–Hadas–Makar-Limanov and quoted from Crachiola) to the filtration of Lemma 3.4 and then invoking Proposition 3.3. Proposition 3.3 is an internal argument using degree filtrations, leading-term comparisons, and an external Epimorphism Theorem (Abhyankar–Moh, Theorem 2.10). None of these reduce to the conclusion: the invariant ring K[x] is not inserted as an assumption, and Lemma 3.2 independently constructs a nontrivial exponential map with invariant ring K[x] to make the ML and DK conclusions two-sided. The possible unstated grading of T in Proposition 3.3 is a proof gap, not circularity, because the conclusion is not fed back as a hypothesis and the cited homogenization theorem is external, not a self-citation. Section 4's isomorphism characterization is derived from the already-established ML/DK computation plus elementary localization and coefficient arguments; it does not assume what it proves. Section 5's stable-isomorphism theorem is a constructive proof that A[1] is isomorphic to B[1] via an explicit exponential map, with the equality E = R^phi justified by Lemma 2.11 and explicit containments; no fitted parameter is renamed as a prediction. Theorem 5.2 combines the non-isomorphism criterion of Theorem 4.1 with the stable isomorphism of Theorem 5.1; pairwise non-isomorphism follows from comparing root multiplicities of X^n g(X), while stable isomorphism follows from the explicit R isomorphic to E[1] construction. The paper contains no self-citation by the author; references to Gupta, Ghosh, Bianchi–Veloso, and Crachiola are external prior work and are not used to forbid alternatives by authorial fiat. Overall, the derivation chain is self-contained against external benchmarks, so the circularity score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central results depend on several substantial external theorems (Abhyankar-Moh, homogenization, standard exponential-map facts) and on the claimed characteristic-free validity of Lemma 3.1. No free parameters or invented entities are present.

assumptions (4)

standard math Theorem 2.8 (homogenization of exponential maps, from Crachiola [11]): any non-trivial exponential map on an admissible filtered domain induces a non-trivial homogeneous exponential map on the associated graded ring with ρ(A^φ) ⊆ gr(A)^φ.
Used in Theorem 3.5 to pass from the surface A to the graded surface B. This is an external theorem cited from the literature.
standard math Lemma 2.4 (standard facts about exponential maps, from [4],[7]): invariant rings are inert, trdeg drops by one, etc.
Used throughout, e.g., to show K[x] is the invariant ring in Lemma 3.2 and to localize in Proposition 3.3.
standard math Theorem 2.10 (Epimorphism theorem of Abhyankar-Moh, from [1],[2]): if K[X,Y]/(h) ≅ K[1] and p∤gcd(deg_X h, deg_Y h), then the degrees divide each other.
Used in Proposition 3.3 to derive a contradiction from the existence of a polynomial-ring structure on a localized surface.
domain assumption Lemma 3.1 (from Bianchi-Veloso [3], Proposition 4): A is a domain, K[x] is inert, localization is K(x)[z], and every element has a unique normal form. The proof is asserted to be characteristic-independent.
The paper relies on these structural properties of the surface A without reproducing the proofs, only stating that the char-0 proof works over any field.

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Pith. "Pith review of On Generalised Danielewski Surfaces over fields of arbitrary characteristic." pith.science (2026). https://pith.science/paper/BD3G6Z3Q

@misc{pith2026250601457,
  author       = {Pith},
  title        = {Pith review of: On Generalised Danielewski Surfaces over fields of arbitrary characteristic},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BD3G6Z3Q}},
  note         = {Machine review of arXiv:2506.01457}
}

abstract

In this paper we study exponential maps ($\mathbb{G}_a$-actions) on the family of affine two dimensional surfaces of the form $f(x)y=\phi(x,z)$ over arbitrary fields, describe the Makar-Limanov invariant and Derksen invariant of these surfaces, give a complete characterization of isomorphisms between such surfaces and display a subfamily which provides counterexamples to the cancellation problem.

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Reference graph

Works this paper leans on

11 extracted references · 11 canonical work pages

[1]

S. S. Abhyankar and T. T. Moh, Embeddings of the line in the plane , J. Reine Angew. Math. 276 (1975), 148–166

work page 1975
[2]

S.S Abhyankar, Lectures on Expansion Techniques In Algebraic Geometry , TIFR(1977)

work page 1977
[3]

A. C. Bianchi and M. O. Veloso, Locally nilpotent derivations and automorphism groups of certain Danielewski surfaces , J. Algebra 469 (2017), 96–108

work page 2017
[4]

Crachiola and L

A. Crachiola and L. Makar-Limanov, On the rigidity of small domains , J. Algebra 284(1) (2005), 1–12

work page 2005
[5]

Derksen, O

H. Derksen, O. Hadas, and L. Makar-Limanov, Newton polytopes of invariants of additive group actions , J. Pure Appl. Algebra 156(2-3) (2001), 187–197

work page 2001
[6]

Ghosh and N

P. Ghosh and N. Gupta, On generalised Danielewski and Asanuma varieties , J. Algebra 632 (2023), 226–250

work page 2023
[7]

Gupta, On the cancellation problem for the affine space A3 in characteristic p , Invent

N. Gupta, On the cancellation problem for the affine space A3 in characteristic p , Invent. Math. 195(1) (2014), 279–288

work page 2014
[8]

Gupta and S

N. Gupta and S. Sen, On double Danielewski surfaces and the cancellation problem, J. Algebra 533 (2019), 25–43

work page 2019

Show all 11 references

[9]

Moser-Jauslin and P

L. Moser-Jauslin and P. M. Poloni, Isomorphisms between cylinders over Danielewski surfaces, Beitr. Algebra Geom. 62(4) (2021), 755– 771

2021
[10]

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

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

1989
[11]

Crachiola, The hypersurface x + x2y + z2 + t3 = 0 over a field of arbitrary characteristic, Proc

A.J. Crachiola, The hypersurface x + x2y + z2 + t3 = 0 over a field of arbitrary characteristic, Proc. Am. Math. Soc. 134 (2005), 1289–1298 16

2005

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

S. S. Abhyankar and T. T. Moh, Embeddings of the line in the plane , J. Reine Angew. Math. 276 (1975), 148–166

work page 1975

[2] [2]

S.S Abhyankar, Lectures on Expansion Techniques In Algebraic Geometry , TIFR(1977)

work page 1977

[3] [3]

A. C. Bianchi and M. O. Veloso, Locally nilpotent derivations and automorphism groups of certain Danielewski surfaces , J. Algebra 469 (2017), 96–108

work page 2017

[4] [4]

Crachiola and L

A. Crachiola and L. Makar-Limanov, On the rigidity of small domains , J. Algebra 284(1) (2005), 1–12

work page 2005

[5] [5]

Derksen, O

H. Derksen, O. Hadas, and L. Makar-Limanov, Newton polytopes of invariants of additive group actions , J. Pure Appl. Algebra 156(2-3) (2001), 187–197

work page 2001

[6] [6]

Ghosh and N

P. Ghosh and N. Gupta, On generalised Danielewski and Asanuma varieties , J. Algebra 632 (2023), 226–250

work page 2023

[7] [7]

Gupta, On the cancellation problem for the affine space A3 in characteristic p , Invent

N. Gupta, On the cancellation problem for the affine space A3 in characteristic p , Invent. Math. 195(1) (2014), 279–288

work page 2014

[8] [8]

Gupta and S

N. Gupta and S. Sen, On double Danielewski surfaces and the cancellation problem, J. Algebra 533 (2019), 25–43

work page 2019