{"id":"f3abd948-7e27-4690-aefa-ce165c674a64","arxiv_id":"2411.14645","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A smooth contractible affine variety with a complexity-two torus action, a unique fixed point, and quotient A2//µ is determined by a linear action on the tangent space and two µ-invariant A1-curves with simple normal crossings.","lead":"The paper classifies smooth, contractible affine spaces with torus group actions of complexity two: each such space is pinned down by a linear action on its tangent space plus two curves in a plane. The result narrows the search for counterexamples to a long-standing linearization conjecture in affine geometry.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 7 Step 1 depends on an unstated fixed-locus connectedness lemma, and local avoidance of D0 alone does not prove the fixed locus is disconnected, so the exclusion of extra divisors is not justified as written.","rationale":"The reader's weakest assumption identifies the same critical step: Step 1 of Theorem 7, where extra prime divisors are excluded. I agree that the unstated connectedness of the fixed locus of a C*-action on a smooth contractible affine variety is load-bearing. However, I also flag a second gap in the same step: even granting the connectedness theorem, the assertion that the T′-fixed locus is disconnected does not follow merely from D0 avoiding an etale neighborhood of q0(x0), because topological connectivity is not a local property and a connected set can meet two such separated regions. This strengthens the need for a citation or an additional argument. I do not see a demonstrated counterexample to the main theorem, and the fixed-locus connectedness statement may well be true in the literature; the paper's examples are consistent with the classification. Therefore I do not change the reader's CONDITIONAL verdict. The proof of Theorem 7 is compressed at a critical point, and a specialist check of Step 1 is required before the classification can be accepted.","tokens_in":11226,"tokens_out":18859,"duration_ms":209795,"concrete_test":"Verify the unstated lemma: for any algebraic C*-action on a smooth contractible affine variety, the fixed-point locus is connected. Attempt a proof via Smith theory and equivariant cohomology localization on the underlying S1-action, recording explicitly where compactness or properness is needed. If the lemma is true, cite it and then re-examine the disconnection step in a concrete AH model, e.g. Y=A2 with an extra divisor D0={v=1} and coefficients as in Example 10; compute the T′-fixed set and check whether it is actually disconnected from the fixed point. If a model exhibits a connected fixed set meeting both x0 and π^{-1}(D0), the proof of Step 1 is invalid.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In the proof of Theorem 7, Step 1, an extra prime divisor D0 with nonempty relative interior coefficient Δ0 is excluded as follows. Choose a one-dimensional subtorus T′ with L∩Δ0 of positive length. Lemma 6 then says every point of X lying over π(D0) is fixed by T′. The proof asserts that because D0 does not meet an etale neighborhood of the image of the unique fixed point x0, the T′-fixed locus is not connected, and this is declared to contradict the unstated fact that the fixed locus of a C*-action on a smooth contractible affine variety is connected. Two load-bearing claims are involved. First, the connectedness theorem is neither stated nor cited; it is nontrivial, since Smith theory gives acyclicity of fixed sets for finite p-groups, and passing to the full C*-fixed locus requires an additional argument. Second, the claimed disconnection is not evident: in an affine variety a connected subvariety can meet both a divisor D0 and a neighborhood of a point y0 even when D0∩U=∅ (for example, a line through 0 and a point at distance 2 in A1). The argument needs a genuine separation statement, e.g. from a Bialynicki-Birula decomposition for the T′-action. If either claim fails, the exclusion of extra divisors collapses, and Theorem 7, hence Theorem 1, are unsupported. The authors should supply a proof or citation of the connectedness lemma and a correct disconnection argument.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a classification of smooth, contractible affine varieties X of dimension n with a faithful algebraic torus action of complexity two, a unique fixed point, and algebraic quotient isomorphic to A2//mu for a finite cyclic group mu (Theorem 1, stated more precisely as Theorem 7). Using the Altmann–Hausen polyhedral divisor framework, the authors claim that such an X is determined by (a) the linear T-action on the tangent space at the fixed point and (b) two mu-invariant curves C1, C2 in A2, each isomorphic to A1 and meeting only with simple normal crossings, whose strict transforms appear as the divisor components D1, D2 in the polyhedral divisor. The paper also states a corollary that every fully hyperbolic complexity-two torus action on affine space is either linearizable or obtained by an equivariant bi-cyclic cover of a linear action, and it gives several examples, including an explicit candidate exotic space.","tokens_in":11498,"tokens_out":4967,"duration_ms":51475,"significance":"If correct, the classification would be a substantial advance on the linearization conjecture for torus actions, reducing the remaining complexity-two case to a very concrete datum: two A1 curves in A2 with SNC crossings. The use of the Altmann–Hausen machinery is appropriate, and the examples are explicit and useful, including a candidate exotic A4 whose Makar-Limanov invariant is reported to be nontrivial. The paper is not machine-checked and contains no parameter fitting; the main result is a structural theorem. However, the proof of the central theorem has at least one load-bearing gap: the exclusion of extra prime divisors in Step 1 of Theorem 7 depends on an unstated and nontrivial connectedness statement for fixed-point loci of C*-actions. Because that step is essential for the claimed classification, the result cannot be considered established as written.","major_comments":[{"comment":"The exclusion of an additional prime divisor D0 relies on two unstated claims. First, the proof asserts that the fixed-point locus of a C*-action on a smooth contractible affine variety must be connected; neither a proof nor a citation is given at the point of use. This is a nontrivial statement (Smith theory gives acyclicity for finite p-groups, and passing to the full C*-fixed locus needs an additional argument). Second, even if connectedness is granted, the proof has not shown that the T'-fixed points lying over pi(D0) are in a different connected component from the fixed point x0. Lemma 6 only describes where T'-fixed points lie; it says nothing about connected components. The statement 'D0 does not intersect an etale neighborhood of the image of x0' does not imply disconnection: in an affine variety a connected subvariety can meet both a divisor D0 and a neighborhood of a point y0 even when D0 avoids that neighborhood, for example a line through 0 and a point at coordinate 2 in A1. The authors need either a proof or citation of the connectedness lemma and a genuine separation argument, for instance from a Bialynicki-Birula decomposition of the T'-action, or a different way to exclude D0. As written, Step 1 and hence Theorem 7 are unsupported.","section":"Section 2, Step 1 of the proof of Theorem 7"},{"comment":"The proof that the curves D1 and D2 are smooth and form an SNC divisor is too compressed. The sentence 'they must be locally isomorphic to toric curves and all toric curves are smooth' is not a proof: it is not established how Proposition 5 gives local isomorphisms of the divisors themselves to toric curves, and the phrase 'toric curves' is not defined in this context. The intended argument presumably uses the etale local description of smooth T-varieties to show that each prime divisor in the polyhedral divisor is a coordinate hyperplane or an exceptional divisor in a linear model, but this is not spelled out. Since Step 2 is the only place where the SNC condition in Theorem 1(b) is justified, it must be expanded.","section":"Section 2, Step 2 of the proof of Theorem 7"},{"comment":"The proof that each Ci is isomorphic to A1 is not sufficiently detailed. The step cites [K93, Theorem A] and asserts that X is obtained by a cyclic cover and is contractible, then claims that the divisors used for the cyclic covering must be Zk-acyclic for almost every k, and that 'Zk-acyclic for almost every k implies that they are, in fact, Zk-acyclic.' Neither the precise statement of [K93, Theorem A] nor the implication 'almost every k' to 'every k' is justified, and the role of contractibility in forcing the branch divisors to have the homology of a point is not explained. Because this is the step that produces the conclusion C1, C2 ≅ A1, the argument needs to be written out with full citations.","section":"Section 2, Step 3(i) of the proof of Theorem 7"}],"minor_comments":[{"comment":"There are several typos and textual artifacts, including 'dimention' in the Introduction, 'strongly convex' misspelled as 'strongly convex' in Section 1.2, and the corrupted reference 'Bia/suppress lynicki-Birula' in the bibliography.","section":"Throughout"},{"comment":"In the proof of Lemma 9, the expression x_i^{a_i q} is used for a rational number q; such an expression is not a well-defined monomial in the coordinate ring. The argument should be formulated using the fact that equality of weight vectors forces the matrix P to have the stated block form without introducing rational powers of coordinates.","section":"Lemma 9"},{"comment":"The displayed algebra C[v, x1, x2, x3, x4]/(x3x4 - x1x3 - v - v^2) appears to contain a typo: substituting u = x1x2 into the two relations x1x2 = u and x3x4 = u + v + v^2 yields the relation x3x4 - x1x2 - v - v^2, not x3x4 - x1x3 - v - v^2.","section":"Example 11"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is in scope and the main theorem, if established, would be a valuable contribution. My recommendation of major revision is driven by the gap in Step 1 of Theorem 7: the unstated connectedness of fixed-point loci and the missing disconnection argument are load-bearing. I would encourage the editor to request a full proof or explicit citation of that lemma and a rewritten Step 1, as well as a more detailed Step 2 and Step 3(i). The paper otherwise appears to have a plausible strategy and useful examples."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Main take: this is a plausible and potentially important classification result in a genuinely hard area, but the proof as written has a real gap in Step 1 of Theorem 7, plus a few other compressed places. It deserves a serious referee, but it is not ready to be accepted as is.\n\nThe genuinely new content is Theorem 7/Theorem 1, extending Koras–Russell and Petitjean's C*-complexity-two results to arbitrary-dimensional torus actions with complexity two. The classification data—the linear action on the tangent space plus two A1 curves in A2/µ with SNC intersections—is natural and the statement is clean. Corollary 8 gives a structural consequence for actions on affine space, and Examples 10–15 illustrate the framework, including an interesting candidate exotic A5 built from the GGP24 preprint. The Altmann–Hausen machinery is appropriate, and the authors' earlier work [LP17] and [P18] is applied as tools, not reverse-engineered. No sign of circularity.\n\nThe soft spots are real. Proposition 3's proof is one sentence; identifying Y as a blow-up of A2/µ from the fan of the quotient toric surface needs more detail. The bigger issue is Step 1 of Theorem 7. The authors exclude an extra divisor D0 by choosing a one-dimensional subtorus T′ and invoking Lemma 6 to get T′-fixed points over π(D0), then say the fixed locus is disconnected because D0 avoids an étale neighborhood of the image of x0, contradicting an unstated claim that the fixed locus of a C*-action on a smooth contractible affine variety is connected. That lemma is not stated or cited, and the disconnection claim does not follow from the avoidance premise alone—a connected set can meet a divisor and a point outside a chosen neighborhood. The stress-test note is right. This step needs either a citation of the connectedness result with correct hypotheses or a genuine separation argument, e.g., via Bialynicki-Birula decompositions. Also, Corollary 8's final sentence says more than the proof establishes, and Theorem 1 is stated without a converse, which is fine but should be explicit. Example 15's exoticness rests on a preprint [GGP24], so treat it as conditional.\n\nBottom line: the main theorem is credible and worth attention, but the write-up is too compressed exactly where it matters. If the authors fill the gap in Step 1 and expand Prop. 3, this becomes a solid paper. I would send it to a knowledgeable referee but expect major revision, not desk rejection.","headline":"Plausible and important classification result, but Step 1 of Theorem 7 has a real gap that needs repair before the paper is publishable.","tokens_in":12047,"tokens_out":3411,"would_cite":true,"duration_ms":34482,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14L30","14R05","14R10","14R20"],"pacs":[],"model":"deepseek-v4-flash","headline":"A smooth, contractible affine variety with a complexity-two torus action, one fixed point, and a two-dimensional quotient is fixed by its tangent-space representation and two affine lines.","keywords":["linearization conjecture","affine varieties","contractible varieties","torus actions","complexity two actions","polyhedral divisors","Altmann-Hausen presentation","exotic affine spaces"],"falsifier":"Find a smooth contractible affine variety with a faithful C*-action whose fixed-point locus has at least two connected components under the surrounding contractibility and quotient assumptions; such an example would collapse the contradiction in Step 1. Alternatively, compute the Altmann--Hausen presentation of any variety satisfying the hypotheses and check whether any prime divisor other than the two curves and the exceptional divisors appears in D; such a divisor would refute the claimed normal form.","tokens_in":2051,"feed_emoji":"📐","tokens_out":2559,"duration_ms":87018,"temperature":0.7,"pith_summary":"The paper proves a classification for the last open class of torus actions relevant to the linearization conjecture: smooth, contractible affine varieties with a faithful complexity-two torus action, exactly one fixed point, and an algebraic quotient isomorphic to the affine plane modulo a finite cyclic group. The claim is that such a variety is completely determined by two pieces of data: the linear action of the torus on the tangent space at the fixed point, and two curves in the quotient plane that are invariant under the cyclic group, pass through the origin, are each isomorphic to the affine line, and meet only with simple normal crossings. This matters because these varieties are the remaining candidates for counterexamples to the linearization conjecture, and the classification reduces the search to a finite combinatorial list while also producing exotic affine spaces.","feed_headline":"Two curves determine smooth contractible torus varieties","feed_subtitle":"A complexity-two torus action with one fixed point is fixed by tangent-space data plus two affine-line curves.","key_machinery":"The load-bearing mechanism is the Altmann--Hausen presentation of affine $T$-varieties by polyhedral divisors: a variety $X$ is encoded as $X(Y,D)$, where $Y$ is a normal semiprojective quotient and $D$ is a formal sum of rational polyhedra supported on prime divisors of $Y$. The key tools are Lemma 6, which converts fixed points of one-dimensional subtori into intervals of positive length in these polyhedra, and Proposition 5, the authors' earlier smoothness criterion, which forces local étale linearity and hence simple normal crossings. The Abhyankar--Moh--Suzuki theorem supplies the final step identifying the two curves as copies of $\\mathbb{A}^1$.","core_discovery":"The central discovery is Theorem 7, stated as Theorem 1 in the introduction: for every smooth, contractible affine variety $X$ with a faithful complexity-two torus action, a unique fixed point $x_0$, and algebraic quotient $\\mathbb{A}^2/\\!/\\mu$, there is an equivariant presentation $X = X(Y,D)$ in the Altmann--Hausen sense, with $D = \\Delta_1\\otimes D_1 + \\Delta_2\\otimes D_2 + \\sum_{i=3}^n \\Delta_i\\otimes E_i$. The coefficients $\\Delta_i$ are computed from the linear $T$-action on the tangent space $T_{x_0}X$; $D_1$ and $D_2$ are the strict transforms of two $\\mu$-invariant curves $C_1,C_2 \\subset \\mathbb{A}^2$ through the origin, each isomorphic to $\\mathbb{A}^1$, intersecting only with simple normal crossings. The proof forces this shape by ruling out extra prime divisors, showing the remaining divisor has simple normal crossings, and applying the Abhyankar--Moh--Suzuki theorem to conclude the curves are affine lines. A corollary is that every fully hyperbolic complexity-two action on the affine space is either linearizable or obtained from a linear action by an equivariant bi-cyclic covering, and the paper exhibits an exotic affine fivefold falling into this classification.","pith_inferences":["Editorial inference: the two-curve data $C_1,C_2$ should make linearizability algorithmically testable: the action is linear precisely when an origin-preserving automorphism of the plane moves both curves to coordinate axes; this follows from Corollary 8 and Proposition 3 but is not stated as a criterion in the paper.","Editorial inference: the same two-curve data may control finer invariants of the resulting variety; computing the Makar-Limanov invariant as a function of $C_1,C_2$ would tell which members of the family are exotic, a computation the paper only performs on one example.","Editorial inference: the classification highlights an unstated topological fact that deserves explicit proof: the fixed-point locus of a $\\mathbb{C}^*$-action on a smooth contractible affine variety is connected. Either this is a known theorem, or the proof of Step 1 needs a replacement argument."],"forward_implications":["If the theorem is correct, the Altmann--Hausen presentation of any such variety is explicitly determined by linear data at the fixed point and by two curves; no other hidden prime divisors can occur.","Corollary 8 follows: every fully hyperbolic complexity-two torus action on affine space is linearizable or an equivariant bi-cyclic cover of a linear action.","The classification produces a smooth, contractible affine fivefold with a unique fixed point that is not isomorphic to affine space (Example 15, with nontrivial Makar-Limanov invariant), so the candidates for linearization counterexamples are genuinely exotic.","Since the exotic fivefold is topologically contractible but not known to be stably affine, the same construction yields potential counterexamples to the Zariski cancellation problem.","The theorem extends the earlier classification of $\\mathbb{C}^*$-actions on contractible threefolds to complexity-two torus actions in all dimensions."],"supporting_citations":[{"why":"Supplies the polyhedral-divisor presentation of affine T-varieties on which the entire proof is built.","marker":"[AH06]"},{"why":"Provides the smoothness criterion forcing local étale linearity and simple normal crossings of the divisor.","marker":"[LP17, Theorem 7]"},{"why":"Gives the fixed-point criterion for one-dimensional torus actions via interval lengths, used in Lemma 6.","marker":"[P18, Proof of Proposition 6]"},{"why":"Supplies the same fixed-point characterization for C*-actions that Lemma 6 extends to subtori.","marker":"[FZ03, Theorem 4.18]"},{"why":"Identifies quotients of fixed-point loci as copies of A1, used in Step 3 to show the curves are affine lines.","marker":"[KPR89, Lemma 5.6]"},{"why":"The Abhyankar--Moh--Suzuki theorem, used to conclude that smooth contractible curves in A2 are isomorphic to A1.","marker":"[AM75, S74]"},{"why":"Fixes the algebraic quotient of Cn by a reductive group as A2//mu, identifying the quotient shape in the hypotheses.","marker":"[GKR08]"},{"why":"Describes cyclic covers of affine T-varieties, used for coefficients with empty relative interior and for the bi-cyclic cover corollary.","marker":"[P15, Lemma 7]"},{"why":"Provides contractibility and Z_k-acyclicity results for cyclic covers used in Step 3 for point coefficients.","marker":"[K93]"},{"why":"The earlier classification of C*-actions on contractible threefolds that this theorem generalizes to complexity two.","marker":"[KR97, Theorem 4.1]"}],"fun_headline_variants":["Two curves fix smooth contractible torus varieties","Complexity-two torus actions: two affine lines are enough","Exotic fivefold emerges from two-curve torus action","Torus actions on affine space: a two-curve classification","Smooth contractible torus varieties decoded by two curves"],"cache_read_input_tokens":14080,"weakest_assumption_plain":"In Step 1 of the proof of Theorem 7, when ruling out an extra prime divisor, the argument relies on the unstated fact that the fixed-point locus of a one-dimensional subtorus action on a smooth contractible affine variety must be connected; if this fact fails, the contradiction does not follow.","fun_headline_variants_meta":{"raw":{"variants":["Two curves fix smooth contractible torus varieties","Complexity-two torus actions: two affine lines are enough","Exotic fivefold emerges from two-curve torus action","Torus actions on affine space: a two-curve classification","Smooth contractible torus varieties decoded by two curves"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000334,"raw_usage":{"total_tokens":1818,"prompt_tokens":873,"completion_tokens":945,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":489,"completion_tokens_details":{"reasoning_tokens":863}},"tokens_in":489,"tokens_out":945,"duration_ms":9215,"temperature":1.0,"reasoning_tokens":863,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T15:03:50.557561+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a smooth contractible affine variety with a faithful C*-action whose fixed-point locus has at least two connected components under the surrounding contractibility and quotient assumptions; such an example would collapse the contradiction in Step 1. Alternatively, compute the Altmann--Hausen presentation of any variety satisfying the hypotheses and check whether any prime divisor other than the two curves and the exceptional divisors appears in D; such a divisor would refute the claimed normal form.","supporting_citations":[],"review_version":1}