A note on smooth $SL_2$-surfaces

Gene Freudenburg

REVIEW 2 major objections 3 minor 37 references

A note on smooth $SL_2$-surfaces

T0 review · 2 major / 3 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A smooth affine SL2-surface X embeds in four-dimensional affine space, yet for every embedding the SL2-action fails to extend, and no embedding into three-dimensional affine space exists.

desk verdict A clean, credible note on the least-studied smooth affine SL2-surface; the non-embedding theorem has one unproved field-extension point that a referee should catch. read the letter →

arxiv 2411.15879 v1 pith:YVDTNJET submitted 2024-11-24 math.AG

Gene Freudenburg This is my paper · ORCID

classification math.AG MSC 13A5014P2514R20

keywords SL2-actionreductivegroupactionML-surfacecancellationproblemlocallynilpotentderivationplinthinvariantDanielewskisurfaceaffine

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 $X$, the quotient of the quadric surface $2x_0x_2 - x_1^2 = 1$ by the sign change in all three coordinates, one of only three smooth affine $SL_2(k)$-surfaces with no nonconstant invertible functions. The main claim is that $X$ behaves differently from the affine plane and the quadric: it embeds into $\mathbb{A}^4_k$, but the $SL_2(k)$-action from its homogeneous-space structure extends to no embedding into $\mathbb{A}^4_k$, and it does not embed into $\mathbb{A}^3_k$ at all. The paper also proves that $X$ is non-cancellative, producing an explicit second surface whose product with the affine line is isomorphic to $X \times \mathbb{A}^1_k$ even though the surfaces themselves are not isomorphic. A reader should care because this yields the first smooth affine $SL_2$-surface whose action cannot be extended for any embedding into the minimal ambient space, and it demonstrates that the plinth invariant can separate such surfaces.

What carries the argument

The load-bearing mechanism is the fundamental pair of locally nilpotent derivations $(\delta, \upsilon)$ on $\mathfrak{R}$ coming from the upper and lower unipotent subgroups of $SL_2(k)$; this pair realizes the $SL_2$-action and induces a $\mathbb{Z}$-grading whose vanishing degrees rule out equivariant embeddings. The second central object is the plinth ideal $\operatorname{pl}(D) = \ker D \cap D\mathfrak{R}$ of an irreducible locally nilpotent derivation; the isomorphism class of $\ker D/\operatorname{pl}(D)$ is the plinth invariant, and the computation $k[x_0^2]/(x_0^2) \cong k$ versus $k[\bar{x}]/(\bar{x}^2)$ is what separates $\mathfrak{R}$ from its cylinder companion $\tilde{\mathfrak{R}}$. The class $\mathcal{C}(k)$ of normal affine surfaces with trivial units, one-dimensional kernels for all nonzero locally nilpotent derivations, and nontrivial Makar-Limanov invariant supplies the setting in which these invariants are defined.

What would settle it

A concrete embedding of $X = \operatorname{Spec}(\mathfrak{R})$ into $\mathbb{A}^3_k$ by polynomial equations would refute Theorem 1.1(d), and an $SL_2(k)$-equivariant embedding into $\mathbb{A}^4_k$, or any algebraic $SL_2(k)$-action on $\mathbb{A}^4_k$ whose restriction to a copy of $X$ is the homogeneous-space action, would refute part (c); since the proof rules out all four linear four-dimensional $SL_2$-modules, such an action would have to be non-linearizable.

Watch

Extended reading notes

Core claim

On its own terms, the paper establishes several structural facts. Theorem 1.1 fixes the embedding behavior of $X = \operatorname{Spec}(\mathfrak{R})$: an algebraic embedding into $\mathbb{A}^4_k$ exists, no equivariant one does, and no algebraic embedding into $\mathbb{A}^3_k$ exists; the equivariant part is proved by checking the four irreducible $SL_2$-modules of dimension four and showing that degree restrictions in the kernel grading collapse their images to $k$, and the $\mathbb{A}^3_k$ part follows from the criterion that a smooth affine surface with trivial Makar-Limanov invariant embeds in $\mathbb{A}^3_k$ only as a Danielewski surface $xy = p(z)$ with simple roots, together with the earlier result that $X$ is not of that form. Theorem 1.2 shows that the automorphism group acts transitively on the irreducible locally nilpotent derivations of $\mathfrak{R}$, with plinth invariant $k$. Theorem 1.3 describes $\operatorname{Aut}_k(\mathfrak{R})$ as $PSL_2(k) \ast_H T$, where $T$ is its triangular subgroup, and shows every automorphism extends to $\mathfrak{D}$. Theorem 1.4 constructs the explicit ring $\tilde{\mathfrak{R}} = \mathfrak{R}[V]/(15x_0^2 V - 3x_0x_1x_2^2 - 2x_1x_2)$ and proves $X \times \mathbb{A}^1_k \cong \tilde{X} \times \mathbb{A}^1_k$ but $X \not\cong \tilde{X}$, while Theorem 1.5 computes plinth invariants $k[x]/(x^n)$ for the Danielewski surfaces $x^n z - y^2 - 1 = 0$.

Load-bearing premise

The non-embedding conclusions rest on two classification results imported without proof: Panyushev's theorem that every algebraic $SL_2(k)$-action on $\mathbb{A}^4_k$ is linear, and the Bandman-Makar-Limanov theorem that a smooth affine surface with trivial Makar-Limanov invariant embeds in $\mathbb{A}^3_k$ only as $xy = p(z)$ with simple roots, with the latter stated for $\mathbb{C}$ and asserted to extend to every algebraically closed field of characteristic zero; the $\mathbb{A}^3_k$ conclusion also uses the earlier result that $X$ is not such a Danielewski surface.

Editorial extensions

If this is right

If Theorem 1.1 is correct, $X$ is the first smooth affine $SL_2$-surface for which the natural action fails to extend to the ambient space for every algebraic embedding into the minimal affine space.
The explicit ring $\tilde{\mathfrak{R}}$ is a concrete cylinder companion to $\mathfrak{R}$: $X \times \mathbb{A}^1_k \cong \tilde{X} \times \mathbb{A}^1_k$ while $X \not\cong \tilde{X}$, giving a new non-cancellative surface.
The plinth invariant computes as $k[x]/(x^n)$ for the surfaces $x^n z - y^2 - 1 = 0$, so it distinguishes all Danielewski surfaces $X_m$ from one another.
The automorphism group formula $\operatorname{Aut}_k(\mathfrak{R}) = PSL_2(k) \ast_H T$ gives a complete description of the symmetry group of $X$ as an amalgamated free product.

Reading between the lines

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

Going beyond the paper: the grading obstruction used to rule out equivariant embeddings may be reusable, since any minimal-dimensional equivariant model of a homogeneous $SL_2$-surface is constrained by the degrees available in a fixed $SL_2$-module, so checking finitely many modules may decide non-extension in other cases.
Going beyond the paper: the plinth pair $(x)$ versus $(x^2)$ suggests that non-cancellation here is detected by the non-reduced fiber over the base point of the quotient map, a local phenomenon that could be engineered in other surfaces with one $\mathbb{G}_a$-fibration.
Going beyond the paper: the open question whether the $SL_2(\mathbb{C})$-action extends holomorphically to $\mathbb{C}^4$ is not settled by these algebraic methods; the behavior of earlier cylinder examples shows holomorphic extension can outrun algebraic extension.

Share X Bluesky LinkedIn Reddit HN

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

This note is worth taking seriously. The surface X is the one smooth affine SL2-surface that had not received a focused treatment; Freudenburg computes real structural information about it: an embedding in A4, non-extension of the action to A4 for every embedding, non-embeddability in A3, the automorphism group as an amalgamated product, and a non-cancellative pair distinguished by the plinth invariant. The non-cancellative example is particularly neat: the plinth invariant distinguishes the two rings after the cylinder is shown to match, and the proof is short and transparent.

The paper is clearly written and honest about its debts. The reductions to Panyushev, Daigle, Wright, and the author's own [17] all look legitimate. I did not find a circularity problem. The self-citations are to published, checkable results.

The main soft spot is Theorem 3.4. Part (d) of Theorem 1.1 depends on an extension of the Bandman-Makar-Limanov theorem from C to every algebraically closed field of characteristic zero, and that extension is asserted in one sentence without proof or reference. This is not a rhetorical quibble: the original proof uses complex topology in essential places, and it is not automatic that the classification transfers. If the author can supply an algebraic proof, or an exact reference where the generalization is proved, then Theorem 1.1(d) is fine. As written, a referee should ask for it. I would not call the whole paper unsound because of this; the rest of the main results stand independently.

Two smaller issues: Theorem 1.5 asks the reader to believe strong transitivity for the B_n's by an "almost identical" argument, but it is not written out; it is easy enough, but should be included. And the proof of Lemma 5.2 leaves several matrix checks to the reader. Both are minor.

All in all, this is a solid subfield-level contribution. For someone working on affine surfaces, complete varieties, or locally nilpotent derivations, it has real content. I would be happy to referee it; I would accept it after the Bandman-Makar-Limanov point is settled. It deserves referee time, not a desk reject.

Referee Report

2 major / 3 minor

Summary. This paper studies the smooth affine SL2-surface X = Spec(R), where R is the ring of invariants of the Z2-action on D = k[x0,x1,x2]/(2x0x2 - x1^2 - 1) sending xi to -xi. The main results are: (1) over algebraically closed characteristic-zero fields, X embeds algebraically into A^4 but not into A^3, and the natural SL2-action on X does not extend to A^4 for any embedding; (2) the automorphism group of R acts transitively on irreducible locally nilpotent derivations, and the plinth invariant of R is k; (3) Aut_k(R) is the amalgamated free product PSL2(k) *H T; (4) R is non-cancellative, witnessed by an explicit overring \tilde{R} such that R^{[1]} ≅ \tilde{R}^{[1]} but R ≇ \tilde{R}; (5) the surfaces Bn defined by x^n z - y^2 -1 = 0 have the strong transitivity property, with plinth invariant k[x]/(x^n).

Significance. If the results are correct, Theorem 1.1 supplies the first smooth affine SL2-surface whose SL2-action is non-extendable for every embedding into the minimal ambient affine space, and the non-cancellative example in Theorem 1.4 is a valuable addition to the cancellation problem literature. The automorphism-group computation and the plinth invariants for the Bn surfaces are also of independent interest. The proofs are mostly detailed and the reductions to established theorems (Panyushev, Bandman–Makar-Limanov, Daigle, Wright) are coherent. The paper would be strengthened by addressing the two gaps noted below, both of which are local and fixable.

major comments (2)

[§3.5, Theorem 3.4] The paper states without proof that the Bandman–Makar-Limanov classification of smooth affine surfaces with ML(S) = C, proved in [3] over C, 'holds more generally for any algebraically closed field k of characteristic zero.' This extension is load-bearing for Theorem 1.1(d), which asserts that X has no algebraic embedding into A^3_k. Since the proof in [3] may use transcendental tools (for instance, the fundamental group at infinity or holomorphic isomorphisms), the transfer to arbitrary algebraically closed fields is not automatic. The author should supply an algebraic proof of this extension or a precise reference; without it, part (d) is not supported for k different from C.
[§8, Theorem 1.5] The claim that each Bn has the strong transitivity property is left to the reader, with the explanation that the proof is 'almost identical' to that of Lemma 5.1. However, Lemma 5.1 relies on Daigle's transitivity theorem for the specific ring D, and that theorem is not directly available for the surfaces Bn (n ≥ 2). Since strong transitivity is a substantive conclusion of Theorem 1.5, a full proof or a specific reference is needed.

minor comments (3)

[§5, Corollary 5.4] In the proof of Corollary 5.4, the phrase 'by Theorem 1.2' should read 'by Theorem 5.3' (or the transitivity result just established), since Theorem 1.2 is the statement being proved.
[Introduction, class C(k)] The paper would be easier to read if the definition of C(k) and the partition C0(k) ∪ C1(k) were accompanied by a one-sentence explanation of why every B ∈ C1(k) has only one kernel; this fact is used in the Introduction without justification.
[§3.3] The elimination of z from the equations of X in A^5 to obtain the A^4 embedding is brief; a short expansion on why the remaining four equations define the same closed subvariety would help the reader verify this step.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central claims are derived from explicit invariant-ring computations and from independent external classification theorems, not from the conclusions themselves.

full rationale

The paper's main results are built from direct computations on the ring R = D^{Z_2}, its fundamental pair of locally nilpotent derivations, explicit generators and relations, the Slice Theorem, and plinth-invariant calculations. Theorem 1.1(b) is an explicit four-generator presentation of R; Theorem 1.1(c) uses Panyushev's classification of SL_2(k)-actions on A^4 together with an explicit computation of the graded pieces of R; Theorem 1.1(d) combines the author's earlier published Proposition 5.3 of [17] with the Bandman-Makar-Limanov theorem. These are external, independently stated results, not reformulations of the target claims. The proof of Theorem 1.4 distinguishes R from tildeR by computing the plinth invariant of R as k and showing tildeR has plinth invariant k[x]/(x^2); this is a genuine invariant computation, not a fitted input or a renamed conclusion. The statement in Section 3.5 that Theorem 3.4 extends from C to every algebraically closed field of characteristic zero is made without proof and is a rigor gap, but it is not circular: the theorem is quoted as an external result and does not encode the non-embedding conclusion. Likewise, reliance on the author's [17] and [16] is self-citation, but these are published, parameter-free results whose assumptions do not include the main conclusions, so they constitute independent support. No step in the derivation chain reduces to its own input, and no prediction is manufactured from fitted data. The appropriate verdict is therefore no significant circularity.

Assumptions & free parameters 0 free parameters · 9 assumptions · 0 invented entities

The central proof imports substantial classification and structure theorems. No fitted parameters are involved. The most license-bound inputs are Panyushev's linearization theorem, the Bandman-Makar-Limanov classification, and previous results of the author for X not being a Danielewski surface. These are accepted as axioms here because they are cited rather than reproved.

assumptions (9)

domain assumption k has characteristic zero; for embedding results k is algebraically closed.
Used throughout; the theorems are stated for characteristic zero, with algebraic closure assumed for Theorem 1.1 and parts of Theorem 1.4(d).
domain assumption Panyushev's theorem: every algebraic SL2(k)-action on A^4 is induced by a linear SL2-module.
Section 4, proof of Theorem 1.1(c); it reduces the case analysis to four 4-dimensional modules.
domain assumption Bandman-Makar-Limanov theorem (Theorem 3.4): for smooth affine surfaces over C with ML(S) = C, embeddability in A^3 is equivalent to being xy = p(t) with simple roots.
Section 3.5 and Section 4(d); the paper extends it to algebraically closed characteristic-zero fields without proof.
domain assumption Popov classification: A^2, Y = SL2/T, and X = SL2/N are the only smooth affine SL2-surfaces with trivial units.
Introduction; frames the paper and identifies X and Y, though the main proofs mostly use the concrete descriptions rather than the full classification.
domain assumption Daigle's transitivity theorem for D and the strong transitivity for D (Lemma 5.1).
Used in the proof of Theorem 5.3 and Theorem 1.2; derived from [6] and Lemma 9.3.
standard math Wright's subgroup theorem for amalgamated free products (Theorem 3.3).
Used in Theorem 1.3(b) to compute Aut_k(R) as an amalgamated free product.
domain assumption Makar-Limanov's computation ML(Bn) = k[x] for Danielewski surfaces with n >= 2.
Used in Theorem 1.5 to identify all kernels of locally nilpotent derivations on Bn.
standard math Slice Theorem and Vasconcelos extension theorem.
Used throughout Sections 3, 5, and 7 for local slices and for extending derivations to integral extensions.
domain assumption Freudenburg [17, Prop. 5.3]: X is not isomorphic to a hypersurface of the form xy = p(z).
Self-cited previous result used in the proof of Theorem 1.1(d).

how reviews work

0 comments

Cite this review

Pith. "Pith review of A note on smooth $SL_2$-surfaces." pith.science (2026). https://pith.science/paper/YVDTNJET

@misc{pith2026241115879,
  author       = {Pith},
  title        = {Pith review of: A note on smooth $SL_2$-surfaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YVDTNJET}},
  note         = {Machine review of arXiv:2411.15879}
}

abstract

Working over a field $k$ of characteristic zero, we study the ring $\mathfrak{R}=\mathfrak{D}^{\mathbb{Z}_2}$ where $\mathfrak{D}=k[x_0,x_1,x_2]/(2x_0x_2-x_1^2-1)$ and $\mathbb{Z}_2$ acts by $x_i\to -x_i$. $\mathfrak{D}$ admits an algebraic $SL_2(k)$-action which restricts to $\mathfrak{R}$. Our results include the following. (1) If $k$ is algebraically closed, the smooth $SL_2$-surface $X={\rm Spec}(\mathfrak{R})$ admits an algebraic embedding in $\mathbb{A}_k^4$, and for any such embedding the $SL_2(k)$-action on $X$ does not extend to $\mathbb{A}_k^4$. In addition, there is no algebraic embedding of $X$ in $\mathbb{A}_k^3$. (2) The automorphism group ${\rm Aut}_k(\mathfrak{R})$ acts transitively on the set of irreducible locally nilpotent derivations of $\mathfrak{R}$. (3) Every automorphism of $\mathfrak{R}$ extends to $\mathfrak{D}$, and ${\rm Aut}_k(\mathfrak{R})=PSL_2(k)\ast_HT$ where $T$ is its triangular subgroup. (4) $\mathfrak{R}$ is non-cancellative, i.e., there exists a ring $\mathfrak{S}$ such that $\mathfrak{R}^{[1]}\cong_k\mathfrak{S}^{[1]}$ but $\mathfrak{R}\not\cong_k\mathfrak{S}$. In order to distinguish $\mathfrak{R}$ from $\mathfrak{S}$, we calculate the plinth invariant for $\mathfrak{R}$.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

37 extracted references · 37 canonical work pages

[17]

Groups 29 (2024), 959–1003

, Actions of SL2(k) on aﬃne k-domains and fundamental pairs , Transform. Groups 29 (2024), 959–1003

work page 2024
[1]

Andrist, J

R. Andrist, J. Draisma, G. Freudenburg, G. Huang, and F. K utzschebauch, A criterion for the algebraic density property of aﬃne SL2-manifolds, (forthcoming)

work page
[3]

Bandman and L

T. Bandman and L. Makar-Limanov, Aﬃne surfaces with AK(S) = C, Michigan Math. J. 49 (2001), 567–582. [4] , Nonstability of AK-invariant , Michigan Math. J. 53 (2005), 263–281

work page 2001
[2]

I. V. Arzhantsev and Y. Zaitseva, Aﬃne homogeneous varieties and suspensions , Res. Math. Sci. 11 (2024), 1–13

work page 2024
[5]

Blanc, A

J. Blanc, A. Bott, and P.-M. Poloni, Real forms of some Gizatullin surfaces and Koras-Russell th reefolds, Publ. Mat. 67 (2023), 851–890

work page 2023
[6]

Daigle, On locally nilpotent derivations of k[x1, x 2, y ]/ (φ (y) − x1x2), J

D. Daigle, On locally nilpotent derivations of k[x1, x 2, y ]/ (φ (y) − x1x2), J. Pure Appl. Algebra 181 (2003), 181–208. A NOTE ON SMOOTH AFFINE SL2-SURF ACES 15

work page 2003
[7]

Daigle and P

D. Daigle and P. Russell, On log Q-homology planes and weighted projective planes , Canad. J. Math. 56 (2004), 1145–1189

work page 2004
[8]

Danielewski, On the cancellation problem and automorphism groups of aﬃne algebraic varieties , Preprint, W arsaw, 1989

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

work page 1989

Show all 37 references

[9]

V. I. Danilov and M. H. Gizatullin, Automorphisms of aﬃne surfaces, II , Math. USSR Izv. 11 (1977), 51–98

1977
[10]

Dubouloz and P.-M

A. Dubouloz and P.-M. Poloni, On a class of Danielewski surfaces in aﬃne 3-space , J. Algebra 321 (2009), 1797–1812

2009
[11]

Ebey, The operation of the universal domain on the plane , Proc

S. Ebey, The operation of the universal domain on the plane , Proc. Amer. Math. Soc. 13 (1962), 722–725

1962
[12]

Fauntleroy and A

A. Fauntleroy and A. Magid, Quasi-aﬃne surfaces with Ga-actions, Proc. Amer. Math. Soc. 68 (1978), 265–270

1978
[13]

Fieseler, On complex aﬃne surfaces with C+-actions, Comment

K.-H. Fieseler, On complex aﬃne surfaces with C+-actions, Comment. Math. Helvetici 69 (1994), 5–27

1994
[14]

Flenner and M

H. Flenner and M. Zaidenberg, Locally nilpotent derivations on aﬃne surfaces with a C∗ -action, Osaka J. Math. 42 (2005), 931–974

2005
[15]

, On the uniqueness of C∗ -actions on aﬃne surfaces , Contemp. Math. 369 (2005), 97–111

2005
[16]

Freudenburg, Algebraic Theory of Locally Nilpotent Derivations , second ed., Encyclopaedia of Mathematical Sciences, vol

G. Freudenburg, Algebraic Theory of Locally Nilpotent Derivations , second ed., Encyclopaedia of Mathematical Sciences, vol. 136, Springer-Verlag, Berlin, Heidelberg, New York, 2017

2017
[18]

Freudenburg and L

G. Freudenburg and L. Moser-Jauslin, Embeddings of Danielewski surfaces , Math. Z. 245 (2003), 823–834

2003
[19]

M. H. Gizatullin, Aﬃne surfaces that are quasihomogeneous wtih respect to an a lgebraic group, Math. USSR Izv. 5 (1971), 754–769

1971
[20]

A. T. Huckleberry, The classiﬁcation of homogeneous surfaces , Expo. Math. 4 (1986), 289–334

1986
[21]

H. W. E. Jung, ¨Uber ganze birationale Transformationen der Ebene , J. Reine Angew. Math. 184 (1942), 161–174

1942
[22]

Kolhatkar, Singular points of aﬃne ML-surfaces , Osaka J

R. Kolhatkar, Singular points of aﬃne ML-surfaces , Osaka J. Math. 48 (2011), 633–644

2011
[23]

Lamy, Sur la structure du groupe d’automorphismes de certaines su rfaces aﬃnes , Publ.Math

S. Lamy, Sur la structure du groupe d’automorphismes de certaines su rfaces aﬃnes , Publ.Math. 49 (2005), 3–20

2005
[24]

Liendo, A

A. Liendo, A. Regeta, and C. Urech, Characterization of aﬃne surfaces with a torus action by the ir automorphism groups, Ann. Sc. Norm. Super. Pias Cl. Sci. 24 (2023), 249–289

2023
[25]

Makar-Limanov, On the group of automorphisms of a class of surfaces , Israel J

L. Makar-Limanov, On the group of automorphisms of a class of surfaces , Israel J. Math. 69 (1990), 250–256

1990
[26]

, On the group of automorphisms of a surface xny = p(z), Israel J. Math. 121 (2001), 113–123

2001
[27]

L. Makar-Limanov, Locally nilpotent derivations on the surface xy = p(z), Proceedings of the Third International Algebra Conference (2002) (Dordrecht, Boston, London), Kl uwer Academic Publishers, 2003, pp. 215–219

2002
[28]

Masuda and M

K. Masuda and M. Miyanishi, The additive group actions on Q-homology planes , Ann. Inst. Fourier (Grenoble) 53 (2003), 429–464

2003
[29]

Nagata, On Automorphism Group of k[x, y ], Lectures in Math

M. Nagata, On Automorphism Group of k[x, y ], Lectures in Math. Kyoto Univ., vol. 5, Kinokuniya Booksto re, Tokyo, 1972

1972
[30]

Panyushev, Semisimple groups of automorphisms of four dimensional spa ce, Math

D.I. Panyushev, Semisimple groups of automorphisms of four dimensional spa ce, Math. USSR-Izv. 23 (1984), 171–183

1984
[31]

V. L. Popov, Classiﬁcation of aﬃne algebraic surfaces that are quasihom ogeneous with respect to an algebraic group, Math. USSR Izv. 7 (1973), 1039–1056

1973
[32]

USSR Izv

, Quasihomogeneous aﬃne algebraic varieties of the group SL( 2), Math. USSR Izv. 7 (1973), 793–831

1973
[33]

Monographs, vol

, Groups, Generators, Syzygies, and Orbits in Invariant Theo ry, Translations of Math. Monographs, vol. 100, Amer. Math. Soc., Providence, 1992

1992
[34]

Rentschler, Op´ erations du groupe additif sur le plan aﬃne , C

R. Rentschler, Op´ erations du groupe additif sur le plan aﬃne , C. R. Acad. Sc. Paris 267 (1968), 384–387

1968
[35]

Serre, Trees, Springer Monographs in Mathematics, Springer-Verlag (Be rlin, Heidelberg, New York), 1980

J.-P. Serre, Trees, Springer Monographs in Mathematics, Springer-Verlag (Be rlin, Heidelberg, New York), 1980

1980
[36]

van der Kulk, On polynomial rings in two variables , Nieuw Arch

W. van der Kulk, On polynomial rings in two variables , Nieuw Arch. Wisk. 1 (1953), 33–41

1953
[37]

W. V. Vasconcelos, Derivations of commutative noetherian rings , Math Z. 112 (1969), 229–233

1969
[38]

D. L. W right, The amalgamated free product structure of GL2([k[X1, ..., X n]) and the weak Jacobian theorem for two variables , J. Pure Appl. Algebra 12 (1978), 235–251. Department of Mathematics Western Michigan University 1903 W. Michigan Ave. Kalamazoo, Michigan 49008 USA ...

1978

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [17]

Groups 29 (2024), 959–1003

, Actions of SL2(k) on aﬃne k-domains and fundamental pairs , Transform. Groups 29 (2024), 959–1003

work page 2024

[2] [1]

Andrist, J

R. Andrist, J. Draisma, G. Freudenburg, G. Huang, and F. K utzschebauch, A criterion for the algebraic density property of aﬃne SL2-manifolds, (forthcoming)

work page

[3] [3]

Bandman and L

T. Bandman and L. Makar-Limanov, Aﬃne surfaces with AK(S) = C, Michigan Math. J. 49 (2001), 567–582. [4] , Nonstability of AK-invariant , Michigan Math. J. 53 (2005), 263–281

work page 2001

[4] [2]

I. V. Arzhantsev and Y. Zaitseva, Aﬃne homogeneous varieties and suspensions , Res. Math. Sci. 11 (2024), 1–13

work page 2024

[5] [5]

Blanc, A

J. Blanc, A. Bott, and P.-M. Poloni, Real forms of some Gizatullin surfaces and Koras-Russell th reefolds, Publ. Mat. 67 (2023), 851–890

work page 2023

[6] [6]

Daigle, On locally nilpotent derivations of k[x1, x 2, y ]/ (φ (y) − x1x2), J

D. Daigle, On locally nilpotent derivations of k[x1, x 2, y ]/ (φ (y) − x1x2), J. Pure Appl. Algebra 181 (2003), 181–208. A NOTE ON SMOOTH AFFINE SL2-SURF ACES 15

work page 2003

[7] [7]

Daigle and P

D. Daigle and P. Russell, On log Q-homology planes and weighted projective planes , Canad. J. Math. 56 (2004), 1145–1189

work page 2004

[8] [8]

Danielewski, On the cancellation problem and automorphism groups of aﬃne algebraic varieties , Preprint, W arsaw, 1989

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

work page 1989