{"id":"505cafea-7b11-4d00-b673-44816a587ced","arxiv_id":"2607.25406","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For anticanonical cones over many Fano varieties, the derived category decomposes as a finite-dimensional algebra component together with two line bundles, and the algebra is explicitly a truncation of a Calabi-Yau completion in the split case.","lead":"This paper constructs explicit categorical absorptions for cone singularities over many Fano varieties, describing the resulting algebras as truncations of Calabi-Yau completions. Read it if you want to see how derived categories of singular spaces can be decomposed into a finite-dimensional algebra piece plus simple geometric pieces, with new equivalences of singularity categories.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The absorption quotient is identified via Proposition 3.9(3.34), a localization equivalence deferred to unpublished [28]; the main algebra descriptions and singular equivalences stand or fall with it.","rationale":"The reader and I identify the same soft spot: Proposition 3.9(3.34). I checked the proof of Proposition 3.9 in the manuscript: the kernel equality (3.33) is proved, but the localization equivalence is explicitly deferred to unpublished work [28]. Corollary 3.11, Corollary 5.10, and Theorem 5.12 all rely on this equivalence to realize the absorption quotient as D_fg(A^•) and to produce the Kawamata-type decomposition D^b(X)=<D^b(A), O_X, O_X(1)>. Without (3.34), the central claim is not fully verified. The paper has honest limitation remarks and a substantial appendix, and I found no internal contradiction; the issue is an external dependency, not a demonstrable error. I therefore recommend keeping the reader's CONDITIONAL verdict unchanged. If the missing lemma appears and is correct, the result likely upgrades; if it fails, the main theorem collapses. The proposed test isolates the localization step from the geometric machinery and would settle whether the deferred lemma is valid in the needed generality.","tokens_in":61347,"tokens_out":5943,"duration_ms":64906,"concrete_test":"Prove (3.34) from (3.33) for the specific non-positive dg algebra \\tilde A^• of Corollary 3.6 (cohomology concentrated in degrees 0 and k+1−d) using only [23, Theorem 1.3] and Lemma 3.10: construct a quasi-inverse to \\bar P_e and check full faithfulness on the t-structure heart. A decisive minimal case is the dg algebra B^• with semisimple part 𝕜×𝕜, one arrow α:1→2 in degree −1, dα=0, α^2=0, and e=e_2; compute whether P_e induces an equivalence D_fg(B^•)/thick(S_1)→D^b(𝕜). If this or the general proof reveals a hidden hypothesis, the main theorem must be revised.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing step is Proposition 3.9(3.34), quoted from the forthcoming Jin–Yang–Zhou [28]. The text proves only the kernel equality (3.33) by a cohomology argument, then states: 'The equivalence (3.34), is shown in [28].' Corollary 3.11 uses this equivalence to identify the Verdier quotient \\tilde A/ker π_* with D_fg(A^•), and Corollary 5.10 / Theorem 5.12 (hence Theorem 1.3) inherit this identification. A functor with a given kernel is not automatically a localization: full faithfulness and essential surjectivity are extra data. Thus the explicit finite-dimensional algebras A, the KSOD D^b(X)=<D^b(A), O_X, O_X(1)>, and the singularity-category equivalences are conditional on an unverified external lemma. This is not a disagreement with consensus; it is an internal completeness gap. The examples and appendix provide supporting evidence, but they do not remove the dependency.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper develops a framework for Kuznetsov–Shinder categorical absorption of isolated cone singularities. For a projective cone X over a strong Fano variety Z (with respect to the anticanonical embedding), the authors prove a semiorthogonal decomposition D^b(X) = <D^b(A), O_X, O_X(1)> with A a finite-dimensional algebra, generalizing earlier nodal cases. The algebra A fits into a square-zero extension 0 → Hom(E, S_E^{-1}E[d-2]) → A → End(E) → 0, and in the split case it is a truncation of Keller's Calabi–Yau completion. The split case is established for many Fano varieties, including projective spaces, smooth quadrics, del Pezzo surfaces of degree > 4, del Pezzo threefolds of degree 5, and finite products thereof. As consequences, the paper obtains singularity-category equivalences D_sg(X) ≅ D_sg(A) and vanishing of K_{-1}(X). An appendix constructs explicit tilting objects on weighted projective spaces P(1^d,m), yielding singular equivalences for cyclic quotient singularities of type (1/m)(1^d).","tokens_in":61624,"tokens_out":8349,"duration_ms":88916,"significance":"If the main theorems hold, they give the first explicit finite-dimensional categorical absorption models for many higher-dimensional cone singularities and provide new Kawamata-type semiorthogonal decompositions in arbitrary dimension. The construction is parameter-free: the bimodule Hom(E, S_E^{-1}E[d-2]) is determined by the geometric exceptional collection, and no fitted constants appear. The paper is technically rich, with detailed proofs, many worked examples, and an independently valuable appendix on tilting objects for weighted projective spaces. However, as detailed below, the central algebra description and the resulting singularity equivalences currently rest on an unproved localization statement quoted from an unpublished preprint, so the overall result is conditional in its present form.","major_comments":[{"comment":"The proof of Proposition 3.9 establishes only the kernel equality (3.33). The crucial equivalence D_fg(B^•)/ker P_e ≅ D_fg(eB^•e) in (3.34) is explicitly stated to be shown in the forthcoming Jin–Yang–Zhou [28]. Corollary 3.11 uses this equivalence to identify the Verdier quotient \\tilde A/ker π_* with D_fg(A^•), and this identification propagates to the main theorems: Theorem 1.3, Theorem 1.8, Corollary 5.10, Theorem 5.12, and Corollary 1.15. A functor with a prescribed kernel is not automatically a localization; full faithfulness and essential surjectivity require proof. The examples and the appendix provide supporting evidence but do not remove this dependency. This is an internal completeness gap, not a disagreement with consensus. I request that the authors either supply a full proof of (3.34) in this paper or explicitly state all theorems depending on it as conditional on [28].","section":"§3.3, Proposition 3.9(3.34)"},{"comment":"The main theorems are stated unconditionally in the introduction, but their proofs depend on Proposition 3.9(3.34), which is quoted from an unpublished preprint. A reader cannot detect this dependency until §3.3. Even if the authors choose to defer the proof to [28], the statements of Theorems 1.3, 1.8, 1.10, and 1.12 should be explicitly qualified as conditional on (3.34), or the proof of (3.34) should be included. As written, the abstract and introduction overstate the current status of the results.","section":"§1.1, Theorems 1.3 and 1.8"}],"minor_comments":[{"comment":"Typographical errors: “equivalances”, “absorbtion”, “satisified”, “ismorphisms”, “correspnding”, “isomoprhism”, and “an an isomorphism” should be corrected. Also “The equivalence (3.34), is shown in [28]” contains an extra comma.","section":"Abstract and throughout"},{"comment":"In the proof of the inclusion “⊆” in (3.33), the text says “Since K P D_fg(eB^•e)”, which appears to be a typo; it should be “K P D_fg(B^•)” to make the argument non-circular.","section":"Proposition 3.9, proof"},{"comment":"The two quivers for Z = P^1 × P^1 are drawn but not clearly labeled; please add vertex labels or describe the quivers in text to make the two choices unambiguous.","section":"Example 1.4(c)"},{"comment":"The symbol E_i is used both for an exceptional sequence and for its endomorphism algebra; please use distinct notation, e.g. E_i for the sequence and E_i^{op} for the algebra, to avoid confusion.","section":"Lemma 4.16"},{"comment":"The paper honestly notes that it is unknown whether non-split deformations occur geometrically. This limitation should be reflected in the introduction’s wording that “in general, the algebras are deformations of the split case”: this is a formal statement (Theorem 4.35), not a claim about geometric examples.","section":"Remark 1.13"},{"comment":"The statement “if m = d, then one can show that the algebra End(⊕_{l=1}^{d-1} G_{l,d,d})^{op} is given by the quiver with relations in Proposition 4.33 and Corollary 4.34” is asserted without proof in the appendix; please indicate where the proof is given or include a reference to the relevant section.","section":"Appendix A.6"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is mathematically substantial and the main construction is elegant. However, the central localization result (Proposition 3.9(3.34)) is quoted from an unpublished preprint, and the main theorems depend on it. I would advise requesting a revision that either proves the lemma or explicitly marks all dependent theorems as conditional. If the authors can include the proof, this is a strong paper; if not, the contribution should be presented as conditional on external work."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nHere is the one-sentence take: this is the first systematic construction of Kawamata-type semiorthogonal decompositions for arbitrary-dimensional anticanonical cone singularities, with explicit absorption algebras in the split case; the main theorems are conditional on a localization lemma the authors defer to an unpublished preprint.\n\nWhat is new and good: the adherence criterion in Section 2, the identification of the split-case absorption algebra as a truncation of Keller's Calabi-Yau completion, the concrete quiver-with-relations descriptions for projective spaces, quadrics, and del Pezzo cases, and the appendix's tilting objects for P(1^d,m). The paper generalizes the Kuznetsov-Shinder nodal results instead of redoing them, and the examples are genuinely informative. The appendix is a substantial piece in itself: it constructs explicit tilting objects for all P(1^d,m) and derives a singular equivalence for P(1^3,2) with the double point ring. The authors are also honest about what they do not know, for instance whether non-split deformations actually arise geometrically.\n\nThe soft spot is exactly where the stress-test note lands. Proposition 3.9(3.34) is the step that passes from the equality of kernels to an equivalence of Verdier quotients. The text proves (3.33) and then says (3.34) is shown in [28]. That is not a small gap in exposition: the presence of a functor with a given kernel is not enough to identify the quotient; you need full faithfulness and essential surjectivity. Every central algebra description and the singularity equivalences in Corollary 3.11, Theorem 5.12, and Corollary 5.10 run through this identification. So as it stands, the paper's main theorem is conditional on an external lemma. The lemma is plausible and probably standard in dg representation theory, and the strategy cited from [65] is reasonable. But \"plausible\" is not a proof, and the referee will need access to [28] before the paper can be fully verified.\n\nThat said, I do not see an internal contradiction, and the missing result is a clearly isolated dependency. The paper is not sloppy; it flags the dependency explicitly. The right response is to send it to a serious referee with instructions to check Proposition 3.9 and to obtain the Jin-Yang-Zhou preprint. If the lemma holds, this is a valuable contribution to the area. If not, the main theorems need repair, but the examples and appendix would still be worth salvaging.\n\nI would take this to reading group and would cite the appendix; I'd accept for peer review, conditional on the external lemma being available and correct.","headline":"Strong conditional contribution: KSODs for cone singularities with explicit absorption algebras, but the central localization lemma is deferred to unpublished Jin-Yang-Zhou, so the main theorems are not fully self-contained.","tokens_in":62081,"tokens_out":3455,"would_cite":true,"duration_ms":40874,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14F08","18G80","14J45","16E45","14B05"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that the derived category of a projective cone over a strong Fano variety has a semiorthogonal decomposition into the derived category of one finite-dimensional algebra and two line bundles, yielding explicit singularity-c","keywords":["categorical absorption","semiorthogonal decomposition","cone singularity","Fano variety","geometric exceptional sequence","Calabi-Yau completion","singularity category","weighted projective space"],"falsifier":"Exhibit a non-positive dg algebra $B$ with finite-dimensional total cohomology and an idempotent $e$ for which the canonical functor $D_{fg}(B)/\\ker P_e \\to D_{fg}(eBe)$ is not an equivalence, while the equality $\\ker P_e = \\operatorname{thick}((H^0(B)/\\bar{e})\\text{-mod})$ holds. Since the paper's Corollary 3.11 and all subsequent algebra descriptions depend on this equivalence, such a counterexample would invalidate the main classification. Alternatively, find a very strong Fano variety $Z$ satisfying Definition 1.1 whose cone yields an algebra $A$ not isomorphic to the split truncation described in Theorem 5.12.","tokens_in":61272,"feed_emoji":"🧩","tokens_out":8298,"duration_ms":76674,"temperature":0.7,"texified_at":"2026-08-05T21:46:48.605075+00:00","pith_summary":"The paper aims to show that for projective cones over a large class of Fano varieties, the bounded derived category admits a semiorthogonal decomposition into a finite-dimensional algebra part and two line bundles, i.e., a Kawamata-type decomposition. In the simplest 'split' case, the algebra is a truncation of a Calabi–Yau completion of the endomorphism algebra of a geometric exceptional sequence; in general, it is a flat deformation of this split algebra. If true, this gives explicit finite-dimensional models for the singularity categories of these cones, hence triangle equivalences between the singularity category of the cone and that of a finite-dimensional algebra, plus vanishing of the first negative K-group. The paper also supplies explicit tilting objects for weighted projective spaces $P(1^d,m)$ in an appendix.","texify_model":"deepseek-v4-flash","texify_usage":{"total_tokens":11423,"prompt_tokens":867,"completion_tokens":10556,"prompt_tokens_details":{"cached_tokens":0},"prompt_cache_hit_tokens":0,"prompt_cache_miss_tokens":867,"completion_tokens_details":{"reasoning_tokens":9686}},"feed_headline":"One algebra captures Fano cone singularities","feed_subtitle":"New semiorthogonal decompositions make singularity categories of Fano cones computable and kill the first negative K-group.","key_machinery":"The central mechanism is the notion of adherence between an exceptional collection $E$ and an autoequivalence $T$ on the crepant resolution, generalizing the one-object case previously studied for nodal singularities. Under adherence, the cones $K_i$ of $E_i \\to T(E_i)$ are $d$-spherical objects generating the kernel of the pushforward, and the absorption algebra is identified as a square-zero extension of $\\operatorname{End}(E)$ by the bimodule $\\operatorname{Hom}(E, S_E^{-1}E[d-2])$ (Proposition 3.8). The identification of the quotient category relies on a localization result for non-positive dg algebras with finite-dimensional cohomology: the kernel of the idempotent functor $P_e$ is $\\operatorname{thick}((H^0(B)/\\bar{e})\\text{-mod})$, and the quotient is $D_{fg}$","core_discovery":"Let $Z$ be a strong Fano variety and $X$ the projective cone over $Z$ via the anticanonical embedding. The main theorem (Corollary 5.10) asserts a semiorthogonal decomposition $D^b(X) = \\langle D^b(A), O_X, O_X(1)\\rangle$ for a finite-dimensional $k$-algebra $A$. When $Z$ is very strong and the exceptional collection consists of sheaves, the algebra $A$ is the split square-zero extension $A \\cong \\operatorname{End}(L) \\oplus \\operatorname{Hom}(L, S_L^{-1}L[d-2])$, which is a truncation of the $(d-1)$-Calabi–Yau completion of $\\operatorname{End}(L)$. In general $A$ is a flat deformation of this split algebra, and the decomposition is admissible with the right orthogonal contained in perfect complexes. The appendix constructs explicit tilting objects $T_d^m$ for weighted projective sp","pith_inferences":["The dependence on the unpublished localization lemma means the algebra identifications are conditional; if the lemma fails, the decompositions may still exist but the explicit A-descriptions would need another proof. An independent verification of the dg-quotient equivalence would settle the main theorem.","Because the split algebras are truncations of Calabi–Yau completions, they are related to higher preprojective algebras; this suggests that the singularity category of these cones might be equivalent to the stable category of a finite-dimensional Frobenius algebra, offering a route to knot-theoretic or mirror-symmetric invariants.","The deformation perspective (flat family connecting absorption algebras) may provide a way to track how categorical absorption varies in families of cones, potentially linking to noncommutative deformations of the underlying Fano variety.","One could test the non-split case by computing Hochschild cohomology H^2(End(L), Hom(L, S^{-1}L[d-2])) for concrete examples; vanishing would imply all cones with the same local singularity are split."],"forward_implications":["For every projective cone over a strong Fano variety, the singularity category D_sg(X) is triangle equivalent to D_sg(A) for an explicit finite-dimensional algebra A; consequently K_{-1}(X) = 0.","The split-case description covers cones over projective spaces, smooth quadrics, del Pezzo surfaces of degree > 4, smooth del Pezzo threefolds of degree five, and finite products; for these, A is a quiver algebra with relations derived from a potential, so the absorption is computable.","When the exceptional collection is not of the required strong type, the paper still obtains a dg-algebra model with cohomology in degrees 0 and k+1−d, yielding a silting object in the perfect subcategory.","The appendix gives explicit tilting objects for weighted projective spaces P(1^d,m) for all m,d>1, including non-Gorenstein cases, and derives singular equivalences, e.g., D_sg(P(1^3,2)) ≅ D_sg(k[z1,z2,z3]/(z1,z2,z3)^2).","Algebras arising from different cones with the same complete local singularity are flat deformations of each other; for endomorphism algebras of global dimension ≤1 only the split algebra occurs."],"fun_headline_variants":["Explicit algebras tame Fano cone singularities","Fano cone singularities reduced to finite algebras","Categorical absorption: split algebras and deformations","One tilting object computes cone singularity categories","New semiorthogonal decompositions for Fano cones"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing premise is a localization equivalence, attributed in the paper to a forthcoming work, that identifies the quotient $D_{fg}(B)/\\ker P_e$ with $D_{fg}(eBe)$ for a non-positive dg algebra $B$ with finite-dimensional cohomology and idempotent $e$; the paper proves only the equality of kernels, and all algebra descriptions rely on this equivalence.","fun_headline_variants_meta":{"raw":{"variants":["Explicit algebras tame Fano cone singularities","Fano cone singularities reduced to finite algebras","Categorical absorption: split algebras and deformations","One tilting object computes cone singularity categories","New semiorthogonal decompositions for Fano cones"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00017,"raw_usage":{"total_tokens":1115,"prompt_tokens":763,"completion_tokens":352,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":507,"completion_tokens_details":{"reasoning_tokens":280}},"tokens_in":507,"tokens_out":352,"duration_ms":4163,"temperature":1.0,"reasoning_tokens":280,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T02:30:41.658634+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Exhibit a non-positive dg algebra $B$ with finite-dimensional total cohomology and an idempotent $e$ for which the canonical functor $D_{fg}(B)/\\ker P_e \\to D_{fg}(eBe)$ is not an equivalence, while the equality $\\ker P_e = \\operatorname{thick}((H^0(B)/\\bar{e})\\text{-mod})$ holds. Since the paper's Corollary 3.11 and all subsequent algebra descriptions depend on this equivalence, such a counterexample would invalidate the main classification. Alternatively, find a very strong Fano variety $Z$ satisfying Definition 1.1 whose cone yields an algebra $A$ not isomorphic to the split truncation described in Theorem 5.12.","supporting_citations":[],"review_version":1}