{"id":"f7793312-e0bc-4f3c-8e79-c66139151e7f","arxiv_id":"2412.18317","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"K-stability is established for infinitely many smooth members of Fano threefold family 2.19, with the main theorem giving a lower bound on local stability thresholds away from one open subset of the exceptional divisor.","lead":"This paper proves K-stability for infinitely many smooth Fano threefolds in the Mukai-Mori family 2.19, the blow-ups of P3 along a smooth genus 2 curve of degree 5 lying on a quadric. For experts in K-stability, it supplies the missing family in a series of blow-up computations and exhibits Abban-Zhuang estimates with weighted blowups.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 4.6, Case 2 uses a non-existent exceptional divisor e5; the resulting S(W)=205/208 is unverified, and the final bound 208/205 leaves only a 3/208 margin.","rationale":"I agree with the reader's weakest-assumption diagnosis. The most load-bearing point is the unverified and internally inconsistent Zariski decomposition in Proposition 4.6, Case 2. The e5 in the u in [1,2] branch is not a matter of taste: no e5 exists in the described blow-up, and the branch transition at v=2-u fails unless e5 is replaced by e2. Because the final delta bound is exactly 1/S(W)=208/205, with S(W)=205/208 only 3/208 away from the dangerous value 1, the computational check is mandatory. The rest of the paper follows standard Abban-Zhuang formalism, and the broad strategy is plausible, but the central theorem cannot be accepted without the corrected computation. This does not change the reader's conditional verdict; it reinforces it.","tokens_in":14802,"tokens_out":8720,"duration_ms":80344,"concrete_test":"Recompute the Zariski decomposition of P(u)|_S - vL in Proposition 4.6 Case 2 using the actual exceptional classes e1 (over p1, with weight (1,2)) and e2, e3, e4, and the corrected third-branch class 2h - e2 - e3 - e4. Substitute into equations (1)-(3) and evaluate the integrals, especially in the region u in [1,2], v in [2-u, 3-3u/2]. If the recomputed S(W) is at least 1, Theorem 1.1 fails in the no-secant case; if the values still match S(W)=205/208 and S(V)=183/208, the e5 is a harmless typo and the proof can be certified after a correction.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Proposition 4.6 Case 2 is the no-secant case needed for Theorem 1.1. The surface S is there described as blowing up p1 with weight (1,2) and p2, p3, p4 ordinarily; no fifth exceptional divisor exists. Nevertheless the displayed Zariski decomposition for u in [1,2] contains the class 2h - e3 - e4 - e5, and the transition at v = 2 - u is inconsistent: the preceding branch evaluates to (2-u)(2h - e2 - e3 - e4), not (2-u)(2h - e3 - e4 - e5). If the intended class is 2h - e2 - e3 - e4, then every subsequent volume integral in equations (1)-(3) must be redone. The paper supplies no intermediate computation for S(V)=183/208 or S(W)=205/208; the final bound delta_p >= 208/205 is the minimum 1/S(W), and S(W)=205/208 is only 3/208 below 1. A corrected decomposition that moves S(W) to or above 1 would destroy the strict inequality and with it Theorem 1.1. This is a computational, not conceptual, weakness, but it is load-bearing because Case 2 covers points through which no secant to C passes.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies K-stability of Fano threefolds obtained by blowing up P3 along a smooth genus 2 curve of degree 5 lying on a smooth quadric, a family that corresponds to rank 2, degree 26 Fano threefolds in the Mori–Mukai list (family 2.19). The main theorem (Theorem 1.1) asserts that the local stability threshold δp(X) is strictly greater than 1 for all points p outside the set E \\ eQ, where E is the exceptional divisor and eQ is the strict transform of the quadric. Using this estimate together with the equivariant K-stability criterion of Zhuang, the authors prove K-stability for infinitely many members of the family (Corollaries 1.2 and 1.3, Example 1.4). The proof is based on the Abban–Zhuang admissible flag technique and consists of a long series of Zariski decomposition and intersection computations.","tokens_in":15053,"tokens_out":8037,"duration_ms":66263,"significance":"If the main theorem is correct, the paper makes a valuable contribution to the classification of K-stable Fano threefolds by providing a unified estimate of the local stability threshold for a previously open family, and it produces an infinite family of explicit K-stable examples. The argument is self-contained modulo standard references and contains no fitted parameters; the numerical outputs (208/205, 52/49, etc.) are fixed intersection-theoretic quantities. The paper also gives a clean criterion based on the absence of fixed points of Aut(P3, C) on C. However, the lack of detail in several key computations and an internal inconsistency in Proposition 4.6 currently prevent full verification of the main claim.","major_comments":[{"comment":"The surface S in Case 2 is obtained from P2 by blowing up p1 with weights (1,2) and p2,p3,p4 ordinarily, so its Picard group has four exceptional divisors: e1 (weighted), e2, e3, e4. Nevertheless, the displayed Zariski decomposition for u ∈ [1,2] contains the classes 2h - e3 - e4 - e5 and (6-3u-2v)(2h - e3 - e4 - e5), involving a fifth exceptional divisor e5 that does not exist in this setup. If the intended class is 2h - e2 - e3 - e4, as the preceding notation suggests, then every volume integral in the definition of S(V) and S(W) must be recomputed; the printed values S(V)=183/208 and S(W)=205/208 are therefore unsupported. The final bound δp ≥ 208/205 leaves a margin of only 3/208, so a corrected computation could easily change the strict inequality. Since Case 2 covers all points through which no secant to C passes, this inconsistency is load-bearing for Theorem 1.1.","section":"Section 4.3, Proposition 4.6, Case 2"},{"comment":"The key estimate 1/S(V^E_{•,•}, Z) = 468n/241 is asserted as a 'direct computation' with no intermediate steps. This lemma is used in the proofs of Corollaries 2.1 and 2.3 to handle divisors with centre a curve in E\\eQ, and without a verifiable computation the reader cannot confirm that the inequality δ_Z(X) > 1 holds. The paper should supply the derivation or at least the explicit volume integrals that lead to this value.","section":"Section 4.1, Lemma 4.2"},{"comment":"The numerical values S(V)=183/208 and S(W)=205/208 in both Case 1 and Case 2 are quoted without showing the actual integrals or the intermediate steps that produce them. In a computation-heavy paper whose final margin is 3/208, such black-box evaluations make it impossible for the referee or reader to check the strict inequality. The authors should include the computed integrals (e.g., a table of contributions per interval in u and v) or provide a reproducible calculation.","section":"Section 4.3, Proposition 4.6"}],"minor_comments":[{"comment":"The quadric equation is written as 'Q = (x0x3 − x1x3 = 0)', which factors as two planes; the correct Segre embedding equation is x0x3 − x1x2 = 0.","section":"Section 2, proof of Corollary 2.3"},{"comment":"The notation for the strict transform of the tangent line is inconsistent: eL, \\tilde L, and \\widetilde L are all used for the same object.","section":"Section 4.3, Proposition 4.6"},{"comment":"In the displayed Zariski decomposition of P(u)|_E − vZ, the interval for v in the second branch is malformed: it appears as 'v ∈ [2−2u/n]' and should be 'v ∈ [0, (2−2u)/n]' (and similarly in the first branch).","section":"Section 4.1, Lemma 4.2"},{"comment":"In Corollary 2.3, the group G is not explicitly identified; the proof appears to use G = Gal(Q̄/Q), but this should be stated clearly when applying [Zhu21, Corollary 4.14].","section":"Section 3.3 and Corollary 2.3"}],"recommendation":"major_revision","confidential_remarks":"The e5 error in Proposition 4.6 looks like a typo, but because the surrounding computation is not shown, it could be either a typo or a genuine mistake. The paper's margin of 3/208 is small, so the authors must fix this and provide the missing computations before the claim can be accepted. If the corrected computation yields S(W) ≥ 1, the main theorem fails; if it still yields S(W) < 1, the paper would be a solid contribution. The authors should also clarify the role of the group G in Corollary 2.3 and fix the typo in the quadric equation. The announced forthcoming paper [Jun] might affect novelty, but that is not a reason to reject the present work if the computations are made complete."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Punchline: this is a genuine new result — K-stability for infinitely many members of family 2.19 — but the written proof is not yet refereed-grade. The main theorem is plausible and the strategy is standard Abban–Zhuang, but one displayed computation contains a clear typo (a non-existent exceptional divisor), and the key numerical values are asserted with no derivation. I think the paper is likely correct, but it needs a computational appendix before anyone should rely on it.\n\nWhat is new and good: the family 2.19 was open; [GGV24] did 2.15 and [CP22] did 2.22, and the paper correctly says [Jun] announces the full family. The infinitely many examples are concrete and the corollaries are a nice payoff. The authors are honest about the limitation of Theorem 1.1 (points in E\\eQ are handled only under an additional group action condition). The external criteria used ([FO18], [BJ20], [AZ22], [Zhu21]) are appropriate. No fitted parameters or circularity.\n\nThe soft spots. Proposition 4.6, Case 2: the hyperplane S meets C in four points, so the blow-up has exceptional divisors e1...e4, not e5. The second displayed branch for u in [1,2] uses 2h-e3-e4-e5; this must be 2h-e2-e3-e4. The two branches also fail to match at v=2-u as written. Because the class is symmetric in the three ordinary exceptions, the stated S(V)=183/208 and S(W)=205/208 might survive the correction, but the paper doesn't show the integration, so the referee has to redo it. Lemma 4.2 similarly states 1/S(V)=468n/241 with no computation. Corollary 2.3's argument that no G-invariant divisors over points in E\\eQ exist is compressed to the point of being unclear; it needs a real proof, not a paragraph. These are fixable, but they are exactly the load-bearing parts of an Abban-Zhuang computation, and the margin in 208/205 is not so large that a small algebraic slip would go unnoticed.\n\nWho this is for: experts in K-stability of Fano threefolds. It deserves a serious referee, but the referee report should be a long list of \"show the computation\" requests. I would not cite it in my own work until the corrected computation appears.","headline":"Genuine new result, plausible proof, but the written computation has a clear typo and several unshown integrals; needs a computational appendix before I would rely on it.","tokens_in":15670,"tokens_out":4052,"would_cite":false,"duration_ms":34218,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14J45","32Q20"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that infinitely many Fano threefolds, obtained by blowing up P^3 along smooth genus-2 degree-5 curves, are K-stable and hence admit Kähler–Einstein metrics.","keywords":["K-stability","stability threshold","delta-invariant","Fano threefolds","blow-up","genus 2 curve","family 2.19","Kähler–Einstein metrics"],"falsifier":"Recompute Proposition 4.6, Case 2 with correct bookkeeping for the hyperplane section: $S\\cap C$ consists of the doubled point $p_1$ plus three further points $p_2,p_3,p_4$, so the displayed class $2h-e_3-e_4-e_5$ should be $2h-e_2-e_3-e_4$. If the resulting $1/S(W_{\\bullet,\\bullet,\\bullet}^{\\widetilde S,\\widetilde L};p)$ is at least $208/205$, the bound survives; if it is smaller, the proof of Theorem 1.1 fails for points with no secant through $C$.","tokens_in":14585,"feed_emoji":"","tokens_out":14974,"duration_ms":115522,"temperature":0.7,"pith_summary":"This paper studies the Fano threefolds obtained by blowing up $\\mathbb P^3$ along a smooth genus-2 curve of degree 5 that lies on a quadric, which is family 2.19 in the standard classification of Fano threefolds. Its main theorem states that for every such blow-up $X$, the local stability threshold satisfies $\\delta_p(X)>1$ for every point $p$ outside the subset $E\\setminus \\widetilde Q$, where $E$ is the exceptional divisor and $\\widetilde Q$ is the strict transform of the quadric. Since a Fano variety is K-stable exactly when its global threshold $\\delta(X)>1$, this local estimate leaves only the curve $E\\setminus \\widetilde Q$ as a possible obstruction. Combining the theorem with an equivariant K-stability criterion, the paper proves K-stability, and hence existence of Kähler–Einstein metrics, for infinitely many explicit members of the family.","feed_headline":"Infinitely many blow-up Fano threefolds are K-stable","feed_subtitle":"Blowing up P3 along a genus-2, degree-5 curve yields varieties that carry Kähler–Einstein metrics.","key_machinery":"The computational engine is the admissible-flag estimate (Theorems 3.7 and 3.8 in [ACC+21]): for a flag $p\\in Z\\subset Y\\subset X$, the local threshold $\\delta_p(X)$ is bounded below by a minimum of three terms involving volumes and orders of vanishing of the positive and negative parts of the Zariski decomposition of $-K_X-uY$. For $p\\in\\widetilde Q$ the flag uses one of the two rulings of the quadric, and at inflexion points a $(1,3)$-weighted blow-up is inserted; for $p\\notin E\\cup\\widetilde Q$ the flag uses the strict transform $\\widetilde S$ of a hyperplane section containing a secant or the tangent line to $C$. The sharp bounds $S_X(\\widetilde Q)=10/13$, $S(V)=183/208$, and the final lower bounds $52/49$ and $208/205$ come from explicit Zariski decompositions and volume integrals.","core_discovery":"The load-bearing result, Theorem 1.1, asserts that for any smooth Fano threefold $X=\\operatorname{Bl}_C\\mathbb P^3$ with $C$ a smooth genus-2, degree-5 curve lying on a smooth quadric, one has $\\delta_p(X)>1$ for all $p\\notin E\\setminus \\widetilde Q$. This is established pointwise: on the strict transform $\\widetilde Q$ of the quadric, Proposition 4.4 gives $\\delta_p(X)\\ge 52/49$; outside $E\\cup\\widetilde Q$, Proposition 4.6 gives $\\delta_p(X)\\ge 208/205$; and for curves in $E$ disjoint from $E\\cap\\widetilde Q$, Lemma 4.2 gives a local bound above 1. The remaining points lie on $E\\setminus\\widetilde Q$, and the paper shows they cannot obstruct K-stability when $C$ satisfies a fixed-point-free automorphism condition (Corollary 1.2) or the rational-roots condition $s_0^2f_3(t_0,t_1)+s_1^2g_3(t_0,t_1)$ with $f_3,g_3$ having no rational solutions (Corollary 1.3). The concrete payoff is an infinite family of explicit curves, for instance $s_0^2(t_0^3-\\eta_0t_1^3)+s_1^2(t_0^3-\\eta_1t_1^3)=0$ with $\\eta_0\\neq\\eta_1$ non-cubes, whose blow-ups are K-stable Fano threefolds in family 2.19.","pith_inferences":["Beyond the paper, if $\\delta_p(X)>1$ could also be proven on $E\\setminus\\widetilde Q$, then every smooth member of family 2.19, not just those with fixed-point-free automorphism action, would be K-stable.","The arithmetic criterion that $f_3$ and $g_3$ have no rational solutions suggests a broader mechanism: for Fano threefolds defined over $\\mathbb Q$, the absence of rational points on the blown-up curve can eliminate equivariant divisors over the exceptional locus and reduce K-stability to pointwise $\\delta$ estimates; the neighbouring families 2.22 and 2.25 are natural test cases.","Because the constants $52/49$ and $208/205$ sit close to 1, an independent computer-algebra check of the volume integrals and Zariski decompositions in Propositions 4.4 and 4.6 would be a decisive verification of the proof's numerical core."],"forward_implications":["For any member of family 2.19 whose automorphism group $\\operatorname{Aut}(\\mathbb P^3,C)$ has no fixed points on $C$, the threefold is K-stable (Corollary 1.2), and because $\\operatorname{Aut}(X)$ is finite this means $X$ admits a Kähler–Einstein metric.","Curves of the form $s_0^2f_3(t_0,t_1)+s_1^2g_3(t_0,t_1)$ with $f_3,g_3$ homogeneous cubics over $\\mathbb Q$ having no rational solutions give K-stable threefolds (Corollary 1.3).","Choosing $\\eta_0\\neq\\eta_1$ non-cube integers in $s_0^2(t_0^3-\\eta_0t_1^3)+s_1^2(t_0^3-\\eta_1t_1^3)=0$ produces infinitely many K-stable examples (Example 1.4).","Pointwise, the paper establishes explicit lower bounds $\\delta_p(X)\\ge 52/49$ on $\\widetilde Q$ and $\\delta_p(X)\\ge 208/205$ outside $E\\cup\\widetilde Q$, so the only possible failure locus for $\\delta>1$ is the curve $E\\setminus\\widetilde Q$."],"supporting_citations":[{"why":"Supplies the admissible-flag framework (Theorems 3.7 and 3.8) that reduces delta_p to multigraded linear series and Zariski decomposition integrals.","marker":"[AZ22]"},{"why":"Provides the book's version of the admissible-flag estimates and the multigraded linear series used throughout Section 4.","marker":"[ACC+21]"},{"why":"Introduces the stability threshold delta whose value above 1 is the paper's working definition of K-stability.","marker":"[FO18]"},{"why":"Establishes the equivalence delta(X)>1 with K-stability and the pointwise reduction delta = inf_p delta_p.","marker":"[BJ20]"},{"why":"Gives the equivariant K-stability criterion (Corollary 4.14) that lets the paper restrict to G-invariant divisors, used in Corollaries 1.2 and 1.3.","marker":"[Zhu21]"},{"why":"Supplies the divisorially K-stable fact S_X(E)<1 used in Lemma 4.2 and in the surface cases.","marker":"[Fuj16]"},{"why":"Classifies which blow-ups of P3 along curves on quadrics are smooth Fano threefolds and gives the bidegree (2,3) and the morphism to the (2,2) complete intersection.","marker":"[BL12]"},{"why":"Shows Aut(X) is finite, which upgrades the K-polystability conclusion to K-stability in Corollary 1.2.","marker":"[CPS19]"}],"fun_headline_variants":["Infinite K-stable Fano threefolds via genus-2 blow-ups","Blow up P^3 along genus-2 curve: K-stable infinitely often","New proof: infinite K-stable blow-up threefolds","Genus-2 curve blow-ups on P^3: infinite K-stable family","P^3 blow-ups along genus-2 curves yield K-stable threefolds"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the Zariski decompositions and the numerical constants derived from them in Propositions 4.4, 4.6, and Lemma 4.2 are computed correctly; if any sharp constant, in particular $208/205$ in Case 2 of Proposition 4.6, is smaller than claimed, the strict inequality $\\delta_p(X)>1$ outside $E\\setminus\\widetilde Q$ can fail and Theorem 1.1 collapses.","fun_headline_variants_meta":{"raw":{"variants":["Infinite K-stable Fano threefolds via genus-2 blow-ups","Blow up P^3 along genus-2 curve: K-stable infinitely often","New proof: infinite K-stable blow-up threefolds","Genus-2 curve blow-ups on P^3: infinite K-stable family","P^3 blow-ups along genus-2 curves yield K-stable threefolds"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000722,"raw_usage":{"total_tokens":3223,"prompt_tokens":915,"completion_tokens":2308,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":531,"completion_tokens_details":{"reasoning_tokens":2204}},"tokens_in":531,"tokens_out":2308,"duration_ms":15001,"temperature":1.0,"reasoning_tokens":2204,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T04:50:28.822909+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Recompute Proposition 4.6, Case 2 with correct bookkeeping for the hyperplane section: $S\\cap C$ consists of the doubled point $p_1$ plus three further points $p_2,p_3,p_4$, so the displayed class $2h-e_3-e_4-e_5$ should be $2h-e_2-e_3-e_4$. If the resulting $1/S(W_{\\bullet,\\bullet,\\bullet}^{\\widetilde S,\\widetilde L};p)$ is at least $208/205$, the bound survives; if it is smaller, the proof of Theorem 1.1 fails for points with no secant through $C$.","supporting_citations":[],"review_version":1}