{"id":"aba5d495-f555-46fd-91e3-75264d9dd40d","arxiv_id":"2507.13649","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"K-stable del Pezzo surfaces with a single quotient singularity of type 1/(mn-1)(1,n) and two exceptional curves in the minimal resolution are exactly S^4_{2,2}, S^5_{2,2}, S^5_{3,2}, S^6_{3,2}, S^6_{4,2}, S^7_{4,2}, S^5_{3,3}, S^6_{4,3}; S^3_{2,2} and S^7_{5,2} are strictly semistable.","lead":"This paper classifies K-stability for del Pezzo surfaces with one quotient singularity whose minimal resolution has two exceptional curves. It identifies the stable surfaces and two special semistable ones, completing the remaining cases in an ongoing classification program.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 3.9's unproved weighted-blow-up identification is load-bearing for the K-stability of S^7_{4,2}, S^6_{4,2}, and S^6_{4,3}.","rationale":"We agree with the reader's diagnosis. Theorem 1.4 is the central claim; its K-stable list includes S^7_{4,2}, S^6_{4,2}, S^6_{4,3}, whose proofs all invoke Lemma 3.9. That lemma's birational identification is stated without proof or reference, and the asserted discrepancy/self-intersection data are not derived. The numerical invariants are consistent (K^2: 4/3 for S versus 1 for S', with E^2 = -3/4 and discrepancy 2/3), which makes the claim plausible; however, consistency does not establish the isomorphism. We also considered the sketchy proof of Aut(S^7_{5,2}) finiteness (Lemma 3.20) and the dimension-count existence of special curves in §3.2.3-3.2.7; both are real gaps but they affect fewer entries or may be repairable, whereas Lemma 3.9 is reused three times and underpins the positive half of the classification. The reader's CONDITIONAL verdict is therefore appropriate: the paper should be accepted only after Lemma 3.9 is supplied with a complete proof or a precise reference.","tokens_in":32462,"tokens_out":20716,"duration_ms":202158,"concrete_test":"Perform an explicit toric computation: represent the 1/3(1,1) singularity of S locally as C^2/Z_3, apply the weighted blow-up with weights (1,4), and compare the resulting surface with S', the blow-up of P(1,1,4) at eight general points. Check that the exceptional divisor E satisfies E^2 = -3/4 and that K_{S'} = f^*K_S + (2/3)E; also compare canonical degree, Picard rank, and singularity type. An independent verification could use the equations: S is a complete intersection of two degree-4 hypersurfaces in P(1,1,2,2,3), and S' is a degree-6 hypersurface in P(1,1,2,3); the weighted blow-up should transform one set of equations into the other.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Lemma 3.9 (smooth-point bound for S^7_{4,2}) asserts that a weighted blow-up with weights (1,4) at the singular point of the surface S in Figure 4 yields exactly the surface S' of Figure 5, with K_{S'} = f^*(K_S) + (2/3)E and E^2 = -3/4. No proof or reference is given for this birational identification or for the discrepancy and self-intersection data. The lemma then uses this identification to transfer the log-canonicity statement of Lemma 3.5 from S to S', and the same transfer is reused for S^6_{4,2} (Lemma 3.16) and S^6_{4,3} (Lemma 3.26). If the weighted blow-up of S at its 1/3(1,1) singular point does not produce the blow-up of P(1,1,4) at eight general points (or produces it with different discrepancy), the contradiction argument in Lemma 3.9 collapses, and the K-stability of three surfaces in the theorem's stable list is unsupported. Because the classification's K-stable list depends on this single geometric fact, it is the most load-bearing step in the paper.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the K-stability of singular del Pezzo surfaces with a single cyclic quotient singularity whose minimal resolution has exactly two exceptional curves with self-intersections -n and -m. For the family S^k_{n,m} constructed by blowing up points on a line in P(1,1,n) and then contracting the strict transform of that line, the authors give a complete classification: eight surfaces are K-stable and two are strictly K-semistable. The proof combines the delta-invariant criterion, Liu's volume bound for K-semistable varieties, and Abban-Zhuang theory, supported by a long sequence of explicit Zariski decompositions and S-invariant computations.","tokens_in":32749,"tokens_out":46478,"duration_ms":493093,"significance":"If the proof is completed, the result gives a full classification of K-(semi)stability for a natural family of singular del Pezzo surfaces with two exceptional curves in the minimal resolution, extending earlier results for blow-ups of P(1,1,n). The computational core is systematic, the arithmetic in the S-invariant calculations is consistent in the cases checked, and the paper makes effective use of external theorems rather than fitted parameters. The main caveats are a missing justification of a key birational identification and a small gap in the transfer argument used for the smooth-point bounds; both appear fixable within the present framework.","major_comments":[{"comment":"The assertion that a weighted blow-up with weights (1,4) at the singular point of S in Figure 4 yields exactly the surface S' of Figure 5, with K_{S'} = f^*K_S + (2/3)E and E^2 = -3/4, is stated without proof or reference. This identification is load-bearing: it transfers the smooth-point log-canonicity statement of Lemma 3.5 to S^7_{4,2}, and it is reused for S^6_{4,2} in Lemma 3.16 and for S^6_{4,3} in Lemma 3.26. Please provide a toric verification using the Hirzebruch-Jung description of the singularity 1/3(1,1), or an explicit reference containing this weighted blow-up computation and the associated discrepancy and self-intersection data.","section":"§3.2.2, Lemma 3.9"},{"comment":"The contradiction to Lemma 3.5 requires the point f(q) to lie in the smooth locus of S. The proof does not rule out the possibility that q lies on the exceptional divisor E of f; in that case f(q) is the singular point of S and Lemma 3.5 does not apply. This gap also affects Lemmas 3.16 and 3.26, which reuse Lemma 3.9. The authors should either justify that q can be chosen away from E (for instance by taking p_1 in the exceptional divisor of π_1 so that q = p ∈ L, which is disjoint from E), or prove the required log canonicity at the singular point of S separately.","section":"§3.2.2, Lemma 3.9"},{"comment":"The proof that Aut(S^7_{5,2}) is finite is too terse. It asserts that the kernel of ρ preserves every geometric basis of Pic(S^7_{5,2}) and that each automorphism in the kernel descends to an automorphism of S^0_{5,2} fixing seven points in general position, but neither statement is justified. This lemma is needed to upgrade δ = 1 to strict K-semistability in Theorem 3.21, so a complete argument or a precise reference is required.","section":"§3.2.5, Lemma 3.20"}],"minor_comments":[{"comment":"There is a typo in the displayed formula: it reads ψ^*(R_S) = π_2^*(C_1 + 4/5 L_1), but the preceding computation gives φ^*(R_{S'}) = C_1 + 5/7 L_1, and the following equality uses C_2 + 5/7 L_2 + 3/7 E. The '4/5' should be '5/7'.","section":"§3.2.2, Lemma 3.8"},{"comment":"The letter S is used for several different surfaces (the blow-up of P(1,1,3) in Figure 4, the blow-up of P(1,1,4) in Figure 5, and a general surface in Lemmas 3.5 and 3.9). Renaming these or adding a table of notation would substantially improve readability.","section":"§3.2.2, Lemma 3.9"},{"comment":"The reference [22, Lemma 4.1] is invoked for log canonicity of a curve C ∈ |O_S(1)|. Since C may be singular and the ambient surface S has a quotient singularity, please state the exact form of the lemma being used or give a short proof of this assertion.","section":"§3.2.1, Lemma 3.5"},{"comment":"The notation O(K^⊥_{S^7_{5,2}}) and the phrase 'geometric basis' are not defined. A more standard formulation of the argument, for example in terms of the action on the Picard lattice and the fixed-point set of the seven marked points, would help the reader.","section":"§3.2.5, Lemma 3.20"}],"recommendation":"major_revision","confidential_remarks":"The unproved weighted blow-up identification in Lemma 3.9 is likely correct and can be verified by a toric computation; the authors should be asked to include it. The paper is well within the scope of the journal and the overall strategy is sound, but the missing justifications are load-bearing for the classification and should be supplied before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my take on the Kim-Lee paper.\n\nThe paper does a real job: it extends the K-stability classification of singular del Pezzo surfaces from the one-exceptional-curve case (Theorem 1.1 of Kim-Won) to the two-exceptional-curve family S^k_{n,m}. The main theorem is a clean list: eight stable surfaces and two strictly semistable ones. The new cases are (3,2), (4,2), (5,2), (3,3), (4,3), and the delta-invariant computations are explicit and consistent. The instability result via Liu's volume bound is straightforward and correct. There's no circularity; the constants are proven bounds, not fitted parameters.\n\nThe soft spot is Lemma 3.9. The authors assert without proof that a weighted blow-up with weights (1,4) at the singular point of S in Figure 4 yields S', the blow-up of P(1,1,4) at eight points, with K_{S'} = f^*K_S + (2/3)E and E^2 = -3/4. This is confusing because S in Figure 4 looks like a smooth blow-up of P(1,1,3) at seven points, not a singular surface. Maybe they mean the singular point of S^6_{3,2} or something else, but as written it's not justified. The same identification is used in Lemma 3.16 and Lemma 3.26 to transfer the log canonicity statement of Lemma 3.5. If this identification fails, the smooth-point bounds for S^7_{4,2}, S^6_{4,2}, and S^6_{4,3} are unsupported, and three entries in the stable list are in question. That's load-bearing.\n\nI checked the surrounding S-invariant computations for arithmetic errors and didn't find any. The Zariski decompositions are laid out carefully. So the gap is specific: it's a missing geometric identification, not a systemic flaw. It could be fixed by a proof or a reference, and the rest of the paper would hold.\n\nThis paper is for the K-stability/classification community. It deserves a serious referee, but the referee should press hard on Lemma 3.9. I'd send it to review and ask the authors to clarify and prove that identification before accepting.\n\nFor a reading group, it's a good paper to discuss because the methods are representative and the gap is instructive. I'd cite the result if I worked in this area, once the gap is resolved.","headline":"Solid computational extension of the K-stability classification, but a load-bearing unproved birational identification in Lemma 3.9 needs to be fixed before the main theorem is fully supported.","tokens_in":33280,"tokens_out":9950,"would_cite":true,"duration_ms":97734,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14J45","14J17","14E30","14J50"],"pacs":[],"model":"deepseek-v4-flash","headline":"For eight singular del Pezzo surfaces with two exceptional curves in the minimal resolution, K-stability holds; two more lie exactly on the semistable boundary.","keywords":["K-stability","del Pezzo surfaces","quotient singularity","delta-invariant","Abban-Zhuang theory","Kähler-Einstein metrics","Zariski decomposition","Fano varieties"],"falsifier":"Write explicit equations for the complete intersection $S$ in $\\mathbb{P}(1,1,2,2,3)$ and the hypersurface $S'$ in $\\mathbb{P}(1,1,2,3)$, perform the weighted blow-up with weights $(1,4)$ at the singular point of $S$, and check whether the resulting surface is isomorphic to $S'$; if it is not, Lemma 3.9 and the stability of $S^7_{4,2}$ would lack support.","tokens_in":1823,"feed_emoji":"📐","tokens_out":2815,"duration_ms":78045,"temperature":0.7,"pith_summary":"This paper aims to complete the K-stability classification for a family of singular del Pezzo surfaces: those with a single cyclic quotient singularity whose minimal resolution has exactly two exceptional curves with self-intersections $-n$ and $-m$. The authors construct these surfaces by blowing up points on weighted projective planes and contracting a curve, producing the family $S^k_{n,m}$. The main theorem states that, within this family, K-stability occurs exactly for eight named surfaces, strict K-semistability for two, and all others are K-unstable. This matters because K-stability is equivalent to the existence of Kähler–Einstein metrics, so the theorem determines exactly which of these surfaces admit such metrics. The proof uses the $\\delta$-invariant and the Abban–Zhuang admissible-flag technique, together with a volume bound that rules out stability in the remaining cases.","feed_headline":"Eight singular del Pezzo surfaces are K-stable, two are borderline","feed_subtitle":"For singular del Pezzo surfaces with a single quotient singularity, K-stability is now fully classified.","key_machinery":"The $\\delta$-invariant stability criterion: a Q-Fano variety is K-semistable if $\\delta(X) \\ge 1$, uniformly K-stable if $\\delta(X) > 1$, and, when the automorphism group is finite, K-stable exactly when K-polystable. The paper estimates local $\\delta$-invariants via the Abban–Zhuang theory of admissible flags, which reduces the computation to S-invariants and explicit Zariski decompositions on weighted blow-ups. A volume bound for quotient singularities, stating that $(-K_X)^n \\le (n+1)^n/|G|$ for K-semistable $X$, is used to prove K-instability for most of the family. For smooth points, the lower bound $\\frac{n+1}{n}\\alpha(X) \\le \\delta(X)$ on the $\\alpha$-invariant is combined with log-canonicity lemmas to show $\\delta > 1$.","core_discovery":"The central claim, Theorem 1.4, is a complete classification of K-stability for the surfaces $S^k_{n,m}$ in Figure 3 and Remark 1.3. If $S$ is a del Pezzo surface in this family, then $S$ is K-stable if and only if it is isomorphic to one of $S^4_{2,2}$, $S^5_{2,2}$, $S^5_{3,2}$, $S^6_{3,2}$, $S^6_{4,2}$, $S^7_{4,2}$, $S^5_{3,3}$, or $S^6_{4,3}$, and $S$ is strictly K-semistable if and only if it is isomorphic to $S^3_{2,2}$ or $S^7_{5,2}$. All other surfaces in the family are K-unstable. For the stable surfaces the local $\\delta$-invariant is shown to exceed $1$; for $S^7_{5,2}$ the $\\delta$-invariant equals $1$ while the automorphism group is finite, which makes it strictly K-semistable but not K-polystable.","pith_inferences":["The same Abban–Zhuang strategy with explicit Zariski decompositions could be applied to quotient singularities with longer Hirzebruch–Jung chains, likely yielding finite classifications for each fixed pair of self-intersections.","The unproved identification in Lemma 3.9, that a weighted blow-up with weights $(1,4)$ at the singular point of the surface in Figure 4 gives the surface in Figure 5, is a concrete computational check; if it fails, the stability of $S^7_{4,2}$ would need a different proof.","The sparsity of stable cases suggests that as $n+m$ grows, the volume bound forces K-instability for almost all members of the family, so the boundary phenomena are concentrated in small $n,m$."],"forward_implications":["The K-stability of every del Pezzo surface in the family $S^k_{n,m}$ is now determined, leaving no unclassified cases.","The eight K-stable surfaces admit Kähler–Einstein metrics by the Yau–Tian–Donaldson correspondence.","The two strictly K-semistable surfaces lie on the boundary of the moduli space and do not admit Kähler–Einstein metrics, despite being semistable.","Because the surfaces are anticanonical models of certain smooth rational surfaces, the K-stability of their minimal resolutions is automatically determined as well.","The result generalizes the earlier one-exceptional-curve classification to the two-exceptional-curve case."],"supporting_citations":[{"why":"Supplies the Abban–Zhuang admissible-flag theory used to estimate local δ-invariants.","marker":"[1]"},{"why":"Defines the δ-invariant and gives the criteria δ ≥ 1 for K-semistability and δ > 1 for uniform K-stability.","marker":"[3]"},{"why":"Provides the volume bound at quotient singularities used to prove K-instability for most surfaces.","marker":"[29]"},{"why":"Gives the result that finite automorphism group implies K-stable if and only if K-polystable, used for the strictly semistable case.","marker":"[2]"},{"why":"Computes δ-invariants of Du Val del Pezzo surfaces of degree at least 4, covering the $S^k_{2,2}$ cases.","marker":"[16]"},{"why":"Computes δ-invariants of cubic surfaces with Du Val singularities, used for $S^3_{2,2}$.","marker":"[17]"},{"why":"Establishes the prior K-stability result for blow-ups of weighted projective planes that this paper generalizes.","marker":"[23]"},{"why":"Shows K-stability transfers between an anticanonical model and its minimal resolution, used to relate the surfaces to their smooth resolutions.","marker":"[40]"}],"fun_headline_variants":["Complete K-stability classification for singular del Pezzo surfaces","Eight stable, two semistable: del Pezzo classification done","Del Pezzo with one quotient singularity: K-stability classified","K-stability fully classified for del Pezzo with one singularity","Eight K-stable del Pezzo surfaces, two borderline: full list"],"cache_read_input_tokens":35456,"weakest_assumption_plain":"The proof that smooth points of $S^7_{4,2}$ satisfy $\\delta > 1$ relies on an asserted isomorphism, stated without proof or reference, between a weighted blow-up with weights $(1,4)$ at the singular point of the surface in Figure 4 and the surface in Figure 5.","fun_headline_variants_meta":{"raw":{"variants":["Complete K-stability classification for singular del Pezzo surfaces","Eight stable, two semistable: del Pezzo classification done","Del Pezzo with one quotient singularity: K-stability classified","K-stability fully classified for del Pezzo with one singularity","Eight K-stable del Pezzo surfaces, two borderline: full list"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001158,"raw_usage":{"total_tokens":4738,"prompt_tokens":831,"completion_tokens":3907,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":447,"completion_tokens_details":{"reasoning_tokens":3819}},"tokens_in":447,"tokens_out":3907,"duration_ms":31744,"temperature":1.0,"reasoning_tokens":3819,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T16:20:13.010874+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Write explicit equations for the complete intersection $S$ in $\\mathbb{P}(1,1,2,2,3)$ and the hypersurface $S'$ in $\\mathbb{P}(1,1,2,3)$, perform the weighted blow-up with weights $(1,4)$ at the singular point of $S$, and check whether the resulting surface is isomorphic to $S'$; if it is not, Lemma 3.9 and the stability of $S^7_{4,2}$ would lack support.","supporting_citations":[{"cited_title":"Abban and Z","cited_arxiv_id":null,"evidence_quote":"Supplies the Abban–Zhuang admissible-flag theory used to estimate local δ-invariants."},{"cited_title":"Blum and M","cited_arxiv_id":null,"evidence_quote":"Defines the δ-invariant and gives the criteria δ ≥ 1 for K-semistability and δ > 1 for uniform K-stability."},{"cited_title":"Araujo, A.-M","cited_arxiv_id":null,"evidence_quote":"Gives the result that finite automorphism group implies K-stable if and only if K-polystable, used for the strictly semistable case."},{"cited_title":"$\\delta$-invariants of Du Val del Pezzo surfaces of degree $\\ge 4$","cited_arxiv_id":"2304.11412","evidence_quote":"Computes δ-invariants of Du Val del Pezzo surfaces of degree at least 4, covering the $S^k_{2,2}$ cases."},{"cited_title":"$\\delta$-invariants of Du Val del Pezzo surfaces of degree $3$","cited_arxiv_id":"2311.14181","evidence_quote":"Computes δ-invariants of cubic surfaces with Du Val singularities, used for $S^3_{2,2}$."},{"cited_title":"Kim and J","cited_arxiv_id":null,"evidence_quote":"Establishes the prior K-stability result for blow-ups of weighted projective planes that this paper generalizes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Shows K-stability transfers between an anticanonical model and its minimal resolution, used to relate the surfaces to their smooth resolutions."}],"review_version":1}