{"id":"baadedf5-8aaf-44eb-8441-6d4dceebbef2","arxiv_id":"2504.15195","paper_version":3,"verdict":"REJECT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"Arc K-semistability is claimed to be a very general algebraic property in flat families via openness of Paul-stability of pairs, but the proof of the pair-stability lemma contains a false assertion about orbit-closure fibers.","lead":"This preprint claims that arc K-semistability and uniform arc K-stability are very general in flat families of polarized varieties. The proof hinges on a new openness theorem for Paul's stability of pairs, but the key geometric lemma in that theorem is false, so the main argument collapses.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 3.1's key identification of the incidence-closure fiber with the orbit closure is false; explicit C* examples show the semistable pair locus is not Zariski open, so Theorem 1.2 and the main results fail.","rationale":"The reader's weakest assumption identifies exactly the same load-bearing error: Proposition 3.1 asserts π^{-1}(y1)=\\overline{G·y1} for the closure of the incidence graph, and this equality is false. My independent counterexample confirms the reader's verdict. The equality fails even for elementary C* actions, and the failure can be realized inside Paul's pair-semistability framework: a fixed semistable point can be a limit of points whose orbit closures meet P(0⊕W). Consequently Corollary 3.2, stated as Zariski openness of semistable pairs, is false, not merely unproved. The reduction in Section 3.2 from arc K-semistability to stability of pairs is interesting and might be salvageable with a different geometric argument or additional hypotheses, but as written the main theorem and its cscK consequences depend on the false openness statement. The paper also omits details in Remark 3.4 and elsewhere, but these omissions are secondary. There is no machine-checked proof or independent verification that would rescue the central claim. The appropriate verdict remains rejection.","tokens_in":11831,"tokens_out":10150,"duration_ms":95229,"concrete_test":"Run the following explicit check. Let G=C*, V=C^2 with weights (0,-1), W=C^2 with weights (0,-1), and set p=[1:0:1:0], q=[1:ε:1:δ] with ε,δ≠0 in P(V⊕W). Verify directly that the orbit of p is {p}, so p is semistable, while lim_{λ→0} λ·q=[0:ε:0:δ]∈P(0⊕W), so q is unstable. This shows the semistable locus is not Zariski open, contradicting Corollary 3.2. Equivalently, test Proposition 3.1 on Y=P^1 with λ·[u:v]=[λu:λ^{-1}v] and Ŵ={[0:1]}: the closure of the incidence graph contains ([1:0],[0:1]), although [0:1] is not in the orbit closure of [1:0], so π^{-1}([1:0])≠\\overline{G·[1:0]}.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central Theorem 1.2 rests entirely on Proposition 3.1, which claims that for the closure Z of the incidence graph of a G-action on Y, the fiber π^{-1}(y) equals the orbit closure \\overline{G·y}. This equality is false. For G=C* acting on P^1 by λ·[u:v]=[λu:λ^{-1}v], take y=[1:0]. The points x_n=[1:1/n] converge to y, and g_n=1/n gives g_n·x_n=[1/n:n]→[0:1]. Hence ([1:0],[0:1]) lies in Z, so π^{-1}([1:0]) contains [0:1], although \\overline{G·[1:0]} is just {[1:0]}. This invalidates the proof that the image of Z∩(Y×Ŵ) under the first projection equals the degenerating locus. The failure is not merely technical: in the pair-semistability setting, take G=C* acting on V⊕W=C^2⊕C^2 with weights 0,-1 on V and 0,-1 on W, and set Ŵ=P(0⊕W). The point p=[1:0:1:0] is fixed and semistable, while for ε,δ≠0 the point q=[1:ε:1:δ] satisfies lim_{λ→0} λ·q=[0:ε:0:δ]∈Ŵ, so q is unstable. Thus every neighbourhood of a semistable point contains unstable points, and the semistable locus is not Zariski open. Since Corollary 3.2, Theorem 1.2, Corollary 3.5, and hence Theorem 1.1 all reduce to this openness assertion, the central claims are unsupported and in fact false.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper aims to prove that arc K-semistability and uniform arc K-stability are very general properties in flat families of polarized varieties (Theorem 1.1). The strategy is to prove that Paul's semistability of pairs is a Zariski open property (Theorem 1.2) via a geometric argument about orbit-closure incidence (Proposition 3.1), then to translate arc K-stability into pair stability using the numerical criteria of Dervan–Reboulet, and finally to apply this to smooth polarized varieties to obtain new examples relevant to constant scalar curvature Kähler metrics.","tokens_in":12158,"tokens_out":5828,"duration_ms":51237,"significance":"If the results were valid, Theorem 1.1 would be a substantial extension of the Blum–Liu–Xu openness results from the Fano case to arbitrary polarized varieties, and the paper would provide a new construction of uniformly arc K-stable varieties. The paper also usefully packages the arc-based numerical criteria from prior work. However, the central geometric lemma is false, and the main theorem is contradicted by an explicit example. Because the claimed Zariski openness of the semistable pair locus fails, the manuscript cannot stand in its present form.","major_comments":[{"comment":"The assertion that the fiber of the closure of the incidence graph equals the orbit closure is false. For G=C* acting on P^1 by λ·[u:v]=[λu:λ^{-1}v], take y0=[1:0], y_j=[1:1/j], and λ_j=1/j. Then λ_j·y_j=[1/j:j]→[0:1], so the pair ([1:0],[0:1]) lies in the closure of the incidence graph. But [0:1] is not in the orbit closure of [1:0], which is just {[1:0]}. The proof's step \"Thus g_j(y1) converges to y2\" is invalid because g_j acts on y1^j, not on y1. This invalidates the identification of the degenerating locus with the image of a closed set.","section":"§3.1, Proposition 3.1"},{"comment":"The claimed Zariski openness of semistable pairs is false. Let G=C* act on V⊕W=C^2⊕C^2 with weights 0,-1 on V and 0,-1 on W, and set Ŵ=P(0⊕W). The point p=[1:0:1:0] is fixed, hence semistable. For any ε,δ≠0, the point q=[1:ε:1:δ] satisfies lim_{λ→0} λ·q=[0:ε:0:δ]∈Ŵ, so q is unstable. Thus every neighborhood of the semistable point p contains unstable points, and the semistable locus is not Zariski open. This directly contradicts Theorem 1.2.","section":"Corollary 3.2 / Theorem 1.2"},{"comment":"The main theorem depends essentially on the false Corollary 3.2 (and on Corollary 3.3, which uses the same openness assertion). Therefore Theorem 1.1, the very generality of arc K-semistability and uniform arc K-stability, is unsupported. Since the counterexample shows the proposed approach cannot work, these claims are not merely unproved but false as stated.","section":"Corollaries 3.5 and 3.6 (Theorem 1.1)"}],"minor_comments":[{"comment":"The notation G·[v:w] in the definition of semistability of pairs should specify whether the orbit or its closure is meant. The proof of Proposition 3.1 uses the closure, and the ambiguity is consequential for the numerical criterion invoked from [DR24].","section":"Definition 2.4"},{"comment":"The supplied text contains numerous character-encoding artifacts (for example \"C/llparenthesist/rrparenthesis\" and similar corrupted sequences). The authors should ensure the final PDF renders all mathematical symbols correctly.","section":"Throughout"},{"comment":"The sentence \"so their complement is a countable union of Zariski closed subsets\" followed by \"the complement of a countable union of countable unions\" is grammatically tangled and should be reworded for clarity.","section":"Proof of Corollary 3.6"}],"recommendation":"reject","confidential_remarks":"The central Proposition 3.1 is a standard pitfall: the closure of the incidence graph can have fibers strictly larger than the orbit closure, and the explicit C* examples in this report disprove Theorem 1.2 and hence Theorem 1.1. The paper's reliance on [DR24] is not itself problematic, but the new geometric step that is supposed to carry the main result fails. In my view, the errors are load-bearing and cannot be repaired within the current framework; a fundamentally different approach would be needed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper sets out to prove that arc K-semistability and uniform arc K-stability are very general in arbitrary flat families of polarized varieties, going well beyond the known Fano case. That is a genuinely important goal, and the idea of reducing it to Paul's stability of pairs is the right kind of move. The paper is also honest about what depends on prior work with Reboulet [DR24]; the numerical criteria are proved there, so the self-citation itself is not the problem.\n\nThe problem is the new geometry. Proposition 3.1, which claims that the fiber of the closure of the incidence graph over y is the orbit closure of y, is false. The proof implicitly assumes that if g_j·y1_j → y2 and y1_j → y1, then g_j·y1 → y2. That is wrong: the group elements can diverge while the moving points compensate. A concrete example is C* acting on P^1 by λ·[u:v]=[λu:λ^{-1}v]. Take y=[1:0] and x_j=[1:1/j], with λ_j=1/j. Then x_j→y, but λ_j·x_j=[1/j:j]→[0:1]. So the pair ([1:0],[0:1]) lies in the closure of the incidence graph, although [0:1] is not in the orbit closure of [1:0]. The equality of fibers is exactly what identifies the image of Z∩(Y×W) with the degenerate locus, so Proposition 3.1 collapses. Consequently Corollary 3.2 and Theorem 1.2, which assert Zariski openness of semistability of pairs, are not proved. In fact they are false: with C* acting on C^2⊕C^2 with weights 0,-1 on both factors, the point [1:0:1:0] is semistable, but every neighborhood contains points [1:ε:1:δ] whose orbit closure meets P(0⊕W). So the semistable locus is not open.\n\nBecause Theorem 1.1 is derived from these statements, the main conclusion is unsupported. There is also a mismatch between the abstract and the body: the abstract claims “first examples of cscK metrics whose existence only follows from the recent solution of Yau–Tian–Donaldson,” while the body says these are varieties not yet known to admit cscK metrics. That should be fixed regardless of the mathematical fate.\n\nThe paper deserves a serious referee because the claim is significant and the error is subtle and instructive. But as written, the central result is false, so my recommendation is reject. If the author can repair Lemma 3.1 or replace it with a different openness argument, the strategy may still yield something real.","headline":"An ambitious and plausible-looking paper whose main theorem rests on a false lemma about incidence graph closures; the result is unproved and the paper should be rejected, though the strategy may be salvageable.","tokens_in":12747,"tokens_out":3916,"would_cite":false,"duration_ms":35032,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14L24","14D20","14J10"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper argues that arc K-semistability is a very general property in flat families of polarized varieties.","keywords":["arc K-semistability","uniform arc K-stability","stability of pairs","Zariski openness","very general property","constant scalar curvature Kähler metrics","flat families of polarized varieties","Hilbert scheme"],"falsifier":"For the $\\mathbb{C}^*$-action $t\\cdot(x,y)=(tx,t^{-1}y)$ on $\\mathbb{A}^2$, compute the fibre of the closure of the graph over the origin: it contains $(0,y)$ for every $y$, although the orbit of the origin is just the origin. That directly tests the equality $\\pi^{-1}(y_1)=\\overline{G\\cdot y_1}$ used in Proposition 3.1.","tokens_in":1764,"feed_emoji":"⚖️","tokens_out":8474,"duration_ms":154423,"temperature":0.7,"pith_summary":"The paper sets out to show that arc K-semistability, a stability condition that tests all formal one-parameter degenerations, behaves algebraically in families. The main theorem claims that in any flat family of polarized varieties the locus of arc K-semistable fibres is very general, meaning its complement is a countable union of Zariski-closed subsets, and the same holds for uniform arc K-stability. The route is to prove a general statement: semistability of a pair in the sense of Paul is Zariski open, and stability of pairs is very general. This matters because it gives the first algebraic very-generality result for these conditions in arbitrary polarized families, and it turns known constant scalar curvature Kähler examples into new uniformly arc K-stable varieties by generic deformation.","feed_headline":"Arc K-semistability is very general in flat families","feed_subtitle":"Stable-pair locus is Zariski open, so cscK existence survives generic deformation.","key_machinery":"The central machinery is stability of pairs: for a group $G$ acting on vector spaces $V$ and $W$, a point $[v:w]\\in\\mathbb{P}(V\\oplus W)$ is semistable when the orbit closure of $[v:w]$ does not meet $\\mathbb{P}(0\\oplus W)$. The paper proves the semistable locus is Zariski open by considering the incidence graph of the group action and its closure in $Y\\times Y$, whose fibres are asserted to be orbit closures; intersecting with $Y\\times\\widehat{W}$ and projecting makes the degeneracy locus closed. A numerical criterion via arcs, formal maps $\\operatorname{Spec}\\mathbb{C}((t))$ to $G$, identifies this condition with the Futaki inequalities defining arc K-semistability. On the Hilbert scheme, the condition is expressed through the CM line bundle written as $L_0 - L_1$, so the argument does not require the CM line bundle to be ample.","core_discovery":"The central claim is that arc K-semistability is very general in flat families of polarized varieties, and uniform arc K-stability is as well. The proof first shows that semistability of a pair is Zariski open, then uses arc-based numerical criteria to translate pair-semistability at a fixed exponent into arc K-semistability at that exponent. This translation takes place on the Hilbert scheme of subvarieties of projective space, where the CM line bundle is expressed as a difference $L_0 - L_1$ of linearised line bundles, so no positivity of the CM line bundle is required. Combining with the known result that smooth polarized varieties with discrete automorphism group and a constant scalar curvature Kähler metric are uniformly arc K-stable, the paper concludes that the constant scalar curvature Kähler locus is very general in flat families, and that generic deformations of finite covers that break covering symmetry give the first examples of uniformly arc K-stable smooth polarized varieties not known to admit such metrics.","pith_inferences":["If the incidence-graph step is correct, the same pair-stability mechanism should give Zariski openness for any stability theory expressed as semistability of a pair, including the mixed Monge-Ampère and harmonic Chern-Weil situations the paper lists.","The Hilbert-scheme formulation covers singular fibres, so the result should feed into moduli constructions for singular polarized varieties once the remaining moduli ingredients such as boundedness and separatedness are supplied.","Because the argument decouples openness from positivity of the CM line bundle, analogous very-generality results should hold for twisted or weighted K-stability conditions.","A useful check is whether the identification of fibres with orbit closures in Proposition 3.1 survives when orbit dimension drops; the $\\mathbb{C}^*$-action $t\\cdot(x,y)=(tx,t^{-1}y)$ on $\\mathbb{A}^2$ is the minimal case to inspect."],"forward_implications":["In any flat family of polarized varieties, arc K-semistability is very general, and uniform arc K-stability behaves the same way.","Semistability of pairs is Zariski open and stability of pairs is very general, supplying the main algebraic ingredient toward Tian's proposed moduli spaces of pairs.","For smooth polarized varieties with discrete automorphism group, the constant scalar curvature Kähler locus is very general in any flat family.","Generic deformations of finite covers of constant scalar curvature Kähler manifolds that break the covering symmetry are uniformly arc K-stable, yielding the paper's advertised examples.","If a fibre admits an arc destabilization whose central fibre is arc K-semistable, then the original fibre must be arc K-semistable."],"supporting_citations":[{"why":"supplies the arc numerical criteria for semistability and stability of pairs and the equivalence with uniform arc K-stability that carries the reduction in Corollary 3.5.","marker":"[DR24]"},{"why":"defines semistability and stability of pairs and supplies the Chow-and-discriminant viewpoint that the present proof generalises through the Hilbert scheme.","marker":"[Pau12]"},{"why":"introduces arc K-semistability and the formalism of models, the notion whose very-generality is the paper's main theorem.","marker":"[Don12]"},{"why":"states the conjecture on quasi-projective moduli of pairs that Theorem 1.2 addresses.","marker":"[Tia13]"},{"why":"provides the determinant expansion used to define the CM line bundle and the weights appearing in the pair construction.","marker":"[KM76]"},{"why":"produces the finite covers with constant scalar curvature Kähler metrics whose generic deformations yield the paper's new uniformly arc K-stable examples.","marker":"[ADVS23]"},{"why":"shows one-parameter subgroups cannot detect pair semistability, motivating why the arc-based numerical criterion is necessary.","marker":"[PSZ24]"}],"fun_headline_variants":["Arc K-stability is very general in flat families","cscK locus is Zariski open in deformations","Uniform arc K-stability survives generic deformations","First cscK metrics via Yau-Tian-Donaldson","Generic flat families admit uniform arc K-stability"],"cache_read_input_tokens":14720,"weakest_assumption_plain":"The load-bearing premise is that the closure of the graph of the group action has exactly the orbit closures as fibres; if a fibre over a degenerate point contains extra points coming from smaller orbits, the Zariski-closedness conclusion can fail.","fun_headline_variants_meta":{"raw":{"variants":["Arc K-stability is very general in flat families","cscK locus is Zariski open in deformations","Uniform arc K-stability survives generic deformations","First cscK metrics via Yau-Tian-Donaldson","Generic flat families admit uniform arc K-stability"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00068,"raw_usage":{"total_tokens":3065,"prompt_tokens":894,"completion_tokens":2171,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":510,"completion_tokens_details":{"reasoning_tokens":2092}},"tokens_in":510,"tokens_out":2171,"duration_ms":14946,"temperature":1.0,"reasoning_tokens":2092,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T11:33:52.202188+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For the $\\mathbb{C}^*$-action $t\\cdot(x,y)=(tx,t^{-1}y)$ on $\\mathbb{A}^2$, compute the fibre of the closure of the graph over the origin: it contains $(0,y)$ for every $y$, although the orbit of the origin is just the origin. That directly tests the equality $\\pi^{-1}(y_1)=\\overline{G\\cdot y_1}$ used in Proposition 3.1.","supporting_citations":[],"review_version":1}