{"id":"60d3054a-c25c-4735-a30f-821151dd0555","arxiv_id":"2411.15879","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The affine SL2-surface X = Spec(D^{Z2}) embeds in A^4 but not A^3, its SL2-action never extends to any A^4 embedding, and it is non-cancellative with automorphism group PSL2(k) *_H T.","lead":"This paper studies a special 2D surface with SL2 symmetries and shows it can be placed in 4D space but not 3D, and its symmetries never extend to the 4D space. It also finds a second surface that becomes identical after adding a dimension, disproving cancellation for this surface.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.1(d) relies on an unproved extension of Bandman–Makar-Limanov's Theorem 3.4 from C to all algebraically closed characteristic-zero fields; the author should supply a proof or reference before acceptance.","rationale":"The central new results are the embedding statements for X and the non-cancellativity example. The explicit construction of X and the proofs of parts (a)–(c) of Theorem 1.1 are self-contained and appear correct; the computations in Sections 3.3–3.4 and the fundamental-pair argument in Section 4 are concrete and checkable. The non-cancellativity proof is also self-contained once the plinth invariant of R is established. My concern is concentrated on Theorem 1.1(d). The proof uses Theorem 3.4, whose statement is imported from [3] over C and then asserted to hold over any algebraically closed field of characteristic zero. This assertion is not proved, and the proof in [3] may rely on complex topology (fundamental groups at infinity, etc.) that does not transfer trivially. Because the theorem is not a single first-order sentence, the Lefschetz principle does not automatically apply. This is a genuine, load-bearing gap in the argument for part (d). The reader's verdict of CONDITIONAL is appropriate; the condition should be that the author either supplies the missing proof of the extension or cites a source proving it over arbitrary algebraically closed fields. The paper's other results do not appear to be affected by this concern.","tokens_in":18664,"tokens_out":19638,"duration_ms":174824,"concrete_test":"Examine [3, Theorems 1–2 and Lemma 5] and identify every step that invokes the complex topology (fundamental group at infinity, homotopy type, holomorphic automorphisms) or the special nature of C. If any such step is essential to the implication (i)⇒(ii) or (i)⇒(iii), then the extension to arbitrary algebraically closed fields is unjustified and the author must provide an algebraic proof or a reference; if the arguments are purely algebraic, the concern is resolved.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Theorem 1.1(d) — no algebraic embedding of X into A^3 — is proved by combining [17, Prop. 5.3] (X is not a Danielewski surface xy=p(t)) with Theorem 3.4, the Bandman–Makar-Limanov classification of smooth affine surfaces S with ML(S)=k. Theorem 3.4 is stated in [3] only over C; the paper asserts, without proof, that it 'holds more generally for any algebraically closed field k of characteristic zero.' This is load-bearing: if the extension fails, part (d) is unsupported for k≠C. The difficulty is that the proof in [3] may use transcendental tools (e.g., the fundamental group at infinity, holomorphic vs algebraic isomorphisms) to prove the equivalence (i) S embeds in A^3 ⇔ (ii) S≅{xy=p(t)} with p simple roots. Such tools do not automatically transfer to arbitrary algebraically closed fields via Lefschetz, since the theorem quantifies over all finitely generated k-algebras and is not a single first-order sentence. The paper gives no algebraic proof or alternative reference.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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).","tokens_in":18833,"tokens_out":11499,"duration_ms":95429,"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":[{"comment":"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.","section":"§3.5, Theorem 3.4"},{"comment":"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.","section":"§8, Theorem 1.5"}],"minor_comments":[{"comment":"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.","section":"§5, Corollary 5.4"},{"comment":"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.","section":"Introduction, class C(k)"},{"comment":"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.","section":"§3.3"}],"recommendation":"major_revision","confidential_remarks":"The central results are credible and the paper is generally well-written. The principal obstacle is the unproved extension of Bandman–Makar-Limanov's theorem to arbitrary algebraically closed fields, which is essential for Theorem 1.1(d). If the author can provide a proof or a reference for this extension, and also complete the proof of strong transitivity for Bn, the paper should be acceptable. The dependence on the author's own [17] and the forthcoming [1] is acceptable but should be verified as available in published form."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nThe 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.\n\nTwo 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.\n\nAll 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.","headline":"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.","tokens_in":19429,"tokens_out":2906,"would_cite":true,"duration_ms":27883,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["13A50","14P25","14R20"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["SL2-action","reductive group action","ML-surface","cancellation problem","locally nilpotent derivation","plinth invariant","Danielewski surface","affine surface"],"falsifier":"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.","tokens_in":18384,"feed_emoji":"📐","tokens_out":10383,"duration_ms":85674,"temperature":0.7,"pith_summary":"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.","feed_headline":"SL2-surface fits in 4-space, but its symmetry never extends","feed_subtitle":"No embedding into A3, no extendable action in any A4 embedding, and a new non-cancellation example.","key_machinery":"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.","core_discovery":"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 $x^n z - y^2 - 1 = 0$.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Popov's classification of smooth affine SL2-surfaces with trivial units identifies X as one of the three cases.","marker":"[31]"},{"why":"Companion classification of quasihomogeneous SL2-surfaces, fixing the central object X.","marker":"[32]"},{"why":"Panyushev's theorem that every algebraic SL2(k)-action on A4 is induced by an SL2-module; load-bearing for Theorem 1.1(c).","marker":"[30]"},{"why":"Bandman-Makar-Limanov criterion that a smooth affine surface with trivial Makar-Limanov invariant embeds in A3 only as xy = p(z) with simple roots; load-bearing for Theorem 1.1(d).","marker":"[3]"},{"why":"Earlier result that X is not isomorphic to xy = p(z); combined with [3] rules out embedding into A3.","marker":"[17]"},{"why":"Provides the defining relations of X in A5 and two locally nilpotent derivations; used for the A4 embedding and later computations.","marker":"[4]"},{"why":"Daigle's transitivity theorem for locally nilpotent derivations of D, the base for the strong transitivity of R.","marker":"[6]"},{"why":"Makar-Limanov's computation ML(Bn) = k[x] for Danielewski surfaces, used in Theorem 1.5.","marker":"[26]"}],"fun_headline_variants":["SL2-surface embeds in A4, but action never extends","No equivariant embedding of SL2-surface in A4","SL2-surface: no A3, no equivariant A4","SL2-surface ring is non-cancellative","SL2-surface: embeds in A4, not in A3"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["SL2-surface embeds in A4, but action never extends","No equivariant embedding of SL2-surface in A4","SL2-surface: no A3, no equivariant A4","SL2-surface ring is non-cancellative","SL2-surface: embeds in A4, not in A3"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001581,"raw_usage":{"total_tokens":6468,"prompt_tokens":1268,"completion_tokens":5200,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":884,"completion_tokens_details":{"reasoning_tokens":5113}},"tokens_in":884,"tokens_out":5200,"duration_ms":36542,"temperature":1.0,"reasoning_tokens":5113,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T13:47:33.568490+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Popov's classification of smooth affine SL2-surfaces with trivial units identifies X as one of the three cases."},{"cited_title":"USSR Izv","cited_arxiv_id":null,"evidence_quote":"Companion classification of quasihomogeneous SL2-surfaces, fixing the central object X."},{"cited_title":"Panyushev, Semisimple groups of automorphisms of four dimensional spa ce, Math","cited_arxiv_id":null,"evidence_quote":"Panyushev's theorem that every algebraic SL2(k)-action on A4 is induced by an SL2-module; load-bearing for Theorem 1.1(c)."},{"cited_title":"Bandman and L","cited_arxiv_id":null,"evidence_quote":"Bandman-Makar-Limanov criterion that a smooth affine surface with trivial Makar-Limanov invariant embeds in A3 only as xy = p(z) with simple roots; load-bearing for Theorem 1.1(d)."},{"cited_title":"Groups 29 (2024), 959–1003","cited_arxiv_id":null,"evidence_quote":"Earlier result that X is not isomorphic to xy = p(z); combined with [3] rules out embedding into A3."},{"cited_title":"Daigle, On locally nilpotent derivations of k[x1, x 2, y ]/ (φ (y) − x1x2), J","cited_arxiv_id":null,"evidence_quote":"Daigle's transitivity theorem for locally nilpotent derivations of D, the base for the strong transitivity of R."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Makar-Limanov's computation ML(Bn) = k[x] for Danielewski surfaces, used in Theorem 1.5."}],"review_version":1}