{"id":"d1551f8b-64b7-41f3-b0bb-747d4192c118","arxiv_id":"2501.10679","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A semi-orthogonal decomposition over the generic fiber of a degeneration need not extend to a decomposition over the whole family, disproving the existence part of the valuative criterion for families of semi-orthogonal decompositions.","lead":"To study shapes called varieties, mathematicians attach a category of algebraic functions and vector bundles. This note shows that a way of splitting such a category on the generic surface of a family can fail to extend to the special central surface, answering an open question in the negative.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 3.5's K0(P_k)=0 step depends on Proposition 3.1 in a regime where the full-exceptional-collection hypothesis is not met; the paper neither states nor proves the needed weak variant.","rationale":"The reader's weakest assumption is exactly the unproved application of Prop 3.1. I agree: this is the load-bearing point because without K0(P_k)=0 the contradiction with the no-phantom central fiber does not go through. The gap is presentational rather than clearly fatal: a mild strengthening of Prop 3.1 likely covers Krah's non-full-but-K0-trivial exceptional collection, and the family over P^1 with central fiber as in [BK24] is a standard degeneration. However, as written, the proof does not contain that strengthening or the required topological-K-theory verification, so a conditional verdict remains appropriate. I would not reject the paper on these grounds, and I would not accept it as complete until the weak variant is supplied.","tokens_in":4534,"tokens_out":16874,"duration_ms":178293,"concrete_test":"Verify the weak form of Proposition 3.1 needed in the proof and check its hypotheses for Krah's phantom: (1) write out the proof of Prop 3.1 with 'full exceptional collection' replaced by 'relative exceptional collection E_1,...,E_n on X_U whose right orthogonal B_U has K0(B_b)=0 and K_top0(B_b)=0 for b in U'; confirm that the two diagrams in the proof still give K0(A_c)≅K0(A_b). (2) For the Krah family over Bo, use the 13 exceptional objects in [Kra23, Thm 1.1] and compute that their classes span K_top0(X_b) integrally, i.e. K_top0(B_b)=0. If (1) and (2) hold, the step is sound; if K_top0(B_b) is nonzero, the argument that K0(P_k)=0 fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Proposition 3.5 asserts: 'We now apply Proposition 3.1 to C and Bo respectively to obtain K0(P_k)=0.' Proposition 3.1, as stated, requires a fiber X_b with a full exceptional collection. The only candidate such fibers in the construction are not identified: general fibers in Bo are blow-ups of P^2 at ten general points with Krah's phantom, and the existence of a full exceptional collection there would force K0 of the phantom component to be a direct summand of a free group, so it cannot be the nonzero phantom of Theorem 3.4. The central fiber X0 is chosen to have no phantom (Theorem 3.6), but the paper never proves it has a full exceptional collection either. What the argument actually needs is a weak form of Proposition 3.1: if over an open U the component B_U is the right orthogonal of a relative exceptional collection E_1,...,E_n with K0(B_b)=0 and K_top0(B_b)=0 for b in U, then K0(A_c)=K0(A_b) for c in an etale neighborhood of b. That weak form is plausible and would fit Krah's construction, but the paper does not state it, and does not verify that Krah's length-13 exceptional collection has trivial right orthogonal in topological K-theory over a relative base. Until this is supplied, K0(P_k)=0 is not justified.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims to disprove the existence part of the valuative criterion for semi-orthogonal decompositions, answering negatively [BOR20, Question 8.8]. The proposed construction is a smooth projective family of rational surfaces over a discrete valuation ring whose generic fiber is a blow-up of the projective plane at ten general points, carrying the Krah phantom as a semi-orthogonal component. The proof shows that this generic decomposition cannot extend to an R-linear decomposition because the central fiber would then carry a phantom, contradicting the Borisov–Kemboi non-existence theorem for blow-ups at points on a smooth cubic. The argument relies on the BOR20 moduli theory of semi-orthogonal decompositions and on Perry's relative topological K-theory.","tokens_in":4840,"tokens_out":12591,"duration_ms":127107,"significance":"If the proof can be made rigorous, the result is a significant and clean counterexample in the deformation theory of semi-orthogonal decompositions, resolving a concrete open question. The paper is commendably short and uses deep external results appropriately, including Krah's phantom and Perry's relative K-theory, and it does not rely on any fitted parameters or ad hoc normalizations. However, as written, the proof contains a load-bearing gap in the step that shows the central fiber component is a phantom; this step is essential for the contradiction. The existence of the family itself is also asserted rather than constructed. With the main gap repaired, the result would be of clear interest to the derived-category and moduli communities.","major_comments":[{"comment":"The proof states: \"We now apply Proposition 3.1 to C and Bo respectively to obtain K0(P_k)=0.\" Proposition 3.1 requires a fiber X_b with a full exceptional collection. The fibers on which the Krah phantom lives are blow-ups of P^2 at ten general points; by Krah's Theorem 3.4, the phantom is the right orthogonal of an exceptional collection of length 13. If that exceptional collection were full, its right orthogonal would be the zero category, contradicting the existence of a nonzero phantom. Thus no such fiber satisfies the hypothesis. The central fiber is only known to have no phantom (Theorem 3.6), not to have a full exceptional collection. Since K0(P_k)=0 is the key step that makes P_k a phantom and produces the contradiction, this gap is load-bearing. The manuscript needs a separate lemma, for instance a weak form of Proposition 3.1: if over an open U the component B_U is the right orthogonal of a relative exceptional collection and K0(B_b)=K_top^0(B_b)=0 for b in U, then K0(P_c)=K0(P_b) for c in an étale neighborhood of b. This variant must be stated and proved, including the topological K-theory comparison for the phantom component, before the argument can proceed.","section":"§3, construction of f"},{"comment":"The proof begins \"we choose a smooth projective morphism f : X → B ≅ P^1 such that...\" but no construction or reference is given showing the existence of a family whose general fibers are blow-ups of P^2 at ten general points and whose special fiber is a blow-up at ten points lying on a smooth cubic. Since Theorem 1.1 is an existence statement, this is part of the required data. A short argument using a rational curve in the Hilbert scheme Hilb^10(P^2) passing through the two specified configurations would fill the gap, but as written the existence of this family is an unproved assumption.","section":"§3, construction of f"},{"comment":"In the commutative diagram argument, the proof claims that because the left and right arrows in the right diagram are injective, the top arrow K0(A_b)→K_top^0(A_b) is an isomorphism. Injectivity of the vertical maps together with an isomorphism on the bottom row does not imply an isomorphism on the top row unless the rows are compatibly split. The desired conclusion does follow from Theorem 2.2(3), which gives explicit direct-sum decompositions of both K0 and K_top^0, but the proof does not state this. The text should be rewritten to use those splittings explicitly; as written, the inference is invalid.","section":"§3, Proposition 3.1"}],"minor_comments":[{"comment":"The manuscript contains several typographical errors in the extracted text, such as \"semi-orth ogonal\", \"d o not\", \"givin g\", and \"Ricolfi\"; these should be corrected in the source file.","section":"Throughout"},{"comment":"The symbol B is used both for the base scheme and for a component of the semi-orthogonal decomposition (e.g., ⟨A, B⟩), and later B_o is an open subset and B′ is a component. This is confusing; renaming the component, for instance to ⟨A, C⟩, would improve readability.","section":"Notation"},{"comment":"The statement \"Since C and SOD_f are both étale over B, we know that C → SOD_f is also étale\" is not a standard cancellation property of étale morphisms and requires either a proof or a reference. If the statement is false in general, the argument should be rephrased to avoid needing it.","section":"§3, proof of Proposition 3.5"}],"recommendation":"major_revision","confidential_remarks":"The manuscript addresses a natural and interesting question, and the overall strategy is promising. The main issue is the unverified application of Proposition 3.1 to the Krah phantom; this is likely repairable with a dedicated lemma, but it is central to the argument. The existence of the family should also be spelled out. I recommend major revision rather than rejection, as no fundamental unsoundness in the central idea is apparent."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this note answers BOR20's Question 8.8 in the negative by combining Krah's phantom on a general fiber with BK24's no-phantom theorem on a special fiber. The construction is clever, the exposition is clean, and the claimed result matters for the moduli theory of semi-orthogonal decompositions. But the proof has a load-bearing gap: Proposition 3.5 applies Proposition 3.1 in a regime where the full exceptional collection hypothesis is not met.\n\nWhat's genuinely new: the counterexample itself. It is a short note built from existing theorems — BOR20's moduli space, Krah's phantom, Perry's relative topological K-theory, BK24's no-phantom result — but the combination is original and answers a real open question. The strategy is sound, and the paper honestly engages with the relevant literature; there is no circularity or fitting.\n\nWhere it gets soft: Proposition 3.5 asserts K0(P_k) = 0 by “applying Proposition 3.1 to C and Bo respectively.” Proposition 3.1 requires a fiber with a full exceptional collection. On the Bo side, fibers are blow-ups of P^2 at ten general points; by Krah's theorem they carry a phantom, so they cannot have a full exceptional collection — a full collection would split K0 and leave no room for a phantom component. On the central side, the special fiber is chosen to have no phantom, but that does not imply it has a full exceptional collection. What the argument actually needs is a weaker statement: if a relative component is the right orthogonal of a relative exceptional collection with trivial K0 on a dense open, then K0 of that component is constant along the family. That statement is plausible, but it is not stated or proved here, and it is not a one-line consequence of Perry's local system. The local system controls topological K-theory, not Grothendieck K0, and the comparison map between them is exactly what can fail in phantom examples.\n\nThere is also a minor sketchiness in the construction of the family: the paper asserts we can choose the degeneration so the central fiber is a blow-up at points on a smooth cubic in very general position. This is plausible and probably fixable, but the referee will want an explicit sentence.\n\nOverall: the result is likely correct, but the write-up is not a complete proof. I would send it to a serious referee because the idea is good and the gap, while substantial, looks fillable. The referee should require a corrected Proposition 3.1 or an alternative argument for constancy of K0 of the phantom component, and should also verify the compatibility of the central fiber with the family.","headline":"Likely-correct negative answer to BOR20's question, but the key K0 comparison in Proposition 3.5 is not proven as written.","tokens_in":5290,"tokens_out":7635,"would_cite":false,"duration_ms":78724,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14F08","14J26"],"pacs":[],"model":"deepseek-v4-flash","headline":"There is a smooth projective family of rational surfaces over a discrete valuation ring whose generic-fiber semi-orthogonal decomposition cannot be extended to the whole family, giving a negative answer to a question from [BOR20].","keywords":["semi-orthogonal decomposition","phantom category","valuative criterion for properness","moduli space of semi-orthogonal decompositions","relative topological K-theory","rational surface","blow-up of the projective plane","family specialization"],"falsifier":"Look at one general fiber of the constructed family and compute the comparison map $K_0(P_b) \\to K^{\\mathrm{top}}_0(P_b/\\mathbb{C})$ for the phantom component; if it is not an isomorphism, the proof's propagation step is not valid, and the claimed contradiction is unsupported.","tokens_in":4330,"feed_emoji":"🧩","tokens_out":7956,"duration_ms":72386,"temperature":0.7,"pith_summary":"This note shows that the existence part of the valuative criterion for properness fails for families of semi-orthogonal decompositions. It constructs a smooth projective family of rational surfaces over a discrete valuation ring such that the derived category of the generic fiber has a semi-orthogonal decomposition that cannot extend to the whole family over the base. Because such extensions are governed by an algebraic moduli space of decompositions, this gives a negative answer to a question posed in [BOR20]. The argument combines the moduli space of semi-orthogonal decompositions with relative topological K-theory and phantom categories.","feed_headline":"Decompositions fail to specialize in a rational-surface family","feed_subtitle":"A phantom on the generic fiber blocks extension, so the moduli space of decompositions is not proper.","key_machinery":"A semi-orthogonal decomposition is an ordered list of full subcategories whose morphisms vanish in one direction and which together generate the derived category. The machinery is the moduli space $\\mathrm{SOD}_f \\to B$ of semi-orthogonal decompositions in a family, the relative topological K-theory functor $K^{\\mathrm{top}}_0(-/B)$, and the notion of a phantom: a nonzero admissible subcategory whose Grothendieck group is zero. The proof positions a phantom from [Kra23] on every general fiber, extends it étale-locally, and uses the local-system property of topological K-theory to force a phantom on the chosen central fiber, where [BK24] says none can exist.","core_discovery":"Theorem 1.1 asserts the existence of a smooth projective family of rational surfaces $f: X \\to \\mathrm{Spec}(R)$ over a discrete valuation ring $R$ and a $K$-linear semi-orthogonal decomposition $\\langle A_K, B_K\\rangle$ of $D^b\\mathrm{Coh}(X_K)$ that is not the base change of any $R$-linear semi-orthogonal decomposition of $D^b\\mathrm{Coh}(X_R)$. This gives a negative answer to Question 8.8 of [BOR20] about the existence part of the valuative criterion for properness. The proof assumes a specialization exists, pulls it back to the closed fiber, shows the relevant component is a phantom, and derives a contradiction from a choice of closed fiber that cannot admit a phantom.","pith_inferences":["A natural next step, not taken in the paper, is to look for an explicit cohomological or deformation-theoretic obstruction class that detects when a generic decomposition cannot specialize; this example shows such an obstruction exists asymptotically but does not name it.","If the K-theory comparison used for phantom components were replaced by another invariant that is deformation-invariant and local in families, the same strategy might produce counterexamples in other settings, such as the K3-category version the author leaves open.","The role of the central fiber is only to be phantom-free; families whose special fiber is a del Pezzo surface might show specialization always holds, drawing a sharper boundary around the failure."],"forward_implications":["The moduli space of semi-orthogonal decompositions of a family is not proper when the base is a curve, since the existence part of the valuative criterion for properness fails.","Question 8.8 in [BOR20] is answered negatively: a decomposition over the generic point cannot always be completed to a decomposition over the whole family.","Phantom components provide an obstruction to specializing semi-orthogonal decompositions, detected through K-theoretic invariants.","The failure already occurs for families of rational surfaces, so it is not a feature of exotic or high-dimensional varieties."],"supporting_citations":[{"why":"Supplies the moduli space of semi-orthogonal decompositions and the question (Question 8.8) that the paper answers negatively; Theorem 2.1 is quoted from here.","marker":"[BOR20]"},{"why":"Constructs a phantom on the blow-up of the projective plane at ten general points, the component used as the obstruction on general fibers.","marker":"[Kra23]"},{"why":"Shows the chosen central fiber (a blow-up at points in very general position on a smooth cubic) admits no phantom, providing the contradiction.","marker":"[BK24]"},{"why":"Supplies the relative topological K-theory functor and its local-system property used to propagate K0 along the family.","marker":"[Per22]"},{"why":"Defines the base change for semi-orthogonal decompositions used to state the specialization condition.","marker":"[Kuz11]"},{"why":"Defines phantom categories as admissible subcategories with trivial K0, the notion at the core of the argument.","marker":"[GO13]"}],"fun_headline_variants":["Semi-orthogonal decompositions fail valuative criterion","Phantom blocks extension of decompositions over DVR","Moduli of decompositions not proper in rational family","No specialization of decompositions across generic fiber"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument needs the K-theory vanishing of the phantom to propagate from a general fiber to the central fiber; this relies on a proposition that assumes the fiber has a full exceptional collection, while the fibers carrying the phantom do not have one, and no variant for this case is supplied.","fun_headline_variants_meta":{"raw":{"variants":["Semi-orthogonal decompositions fail valuative criterion","Phantom blocks extension of decompositions over DVR","Moduli of decompositions not proper in rational family","No specialization of decompositions across generic fiber"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00014,"raw_usage":{"total_tokens":1048,"prompt_tokens":719,"completion_tokens":329,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":335,"completion_tokens_details":{"reasoning_tokens":267}},"tokens_in":335,"tokens_out":329,"duration_ms":3973,"temperature":1.0,"reasoning_tokens":267,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T19:05:24.587242+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Look at one general fiber of the constructed family and compute the comparison map $K_0(P_b) \\to K^{\\mathrm{top}}_0(P_b/\\mathbb{C})$ for the phantom component; if it is not an isomorphism, the proof's propagation step is not valid, and the claimed contradiction is unsupported.","supporting_citations":[],"review_version":1}