{"id":"8d9ca200-d055-4eb3-a16b-f234d3815f51","arxiv_id":"2505.22528","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Exact formulas are derived for the δ-invariant of (P2, λC_d) for all plane curves C_d of degree d ≤ 4, classified by singularity type.","lead":"This paper computes the δ-invariant, a key stability measure, for pairs consisting of the projective plane and a plane curve of degree up to 4. This gives a reference table of K-stability results for many log Fano pairs and provides tools for proving K-stability of higher-dimensional Fano varieties.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The exact δ-values rest on applying Abban–Zhuang's lower bound (2.0.2) to weighted-blowup models with quotient singularities on the exceptional divisor; the required plt/klt/ampleness hypotheses are never verified case by case.","rationale":"The reader's weakest assumption identifies the same place where the argument is least secure: the use of (2.0.2) on singular intermediate surfaces. The proofs construct weighted blowups whose exceptional divisors pass through cyclic quotient singularities of the model, and the hypotheses of the imported theorem are not verified case by case. This is a genuine gap in presentation, but it is likely repairable, since these models are standard plt blowups and the claimed differents have the expected coefficients. No independent arithmetic contradiction was found in the volumes; the many typographical errors (e.g., Lemmas 6.2 and 6.3 both state multiplicity 3, and Theorem 20.2 gives interval [0,1/3] while Lemma 20.1 proves [0,1/2]) affect statements but not the core method. Therefore the conditional verdict is appropriate, and the concrete test above would convert the conditional acceptance into a verified theorem if it succeeds.","tokens_in":57665,"tokens_out":26320,"duration_ms":313234,"concrete_test":"For the model of Lemma 6.1, write S explicitly as the weighted blowup of P2 at P with weights (1,2). Resolve the 1/2(1,1) quotient singularity and compute the different Δ_E from the formula K_S + λC' + E = K_E + Δ_E. Check that (i) E is smooth rational, (ii) -(K_E+Δ_E) is σ|_E-ample for λ∈[0,3/4], and (iii) Δ_E = (1/2)P + λQ is klt. Then recompute S(W^E;O) from its definition on the resolution and compare with the printed bounds (3-4λ)/6 and (3-4λ)/3. If any of (i)–(iii) fails, or if the integrals differ, the equality δ(P2,λC4) = (3/4)(4-3λ)/(3-4λ) in the absence of a 4-tangent is not established.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central theorem asserts exact values of δ, so each proof needs both an upper bound from a divisor E and a matching lower bound. The lower bound is imported from (2.0.2), whose hypotheses are listed in Section 2: the model must be a plt blowup with E smooth rational, the different Δ_E must be klt, and -(K_E+Δ_E) must be ample. In Lemmas 4.1, 5.1, 6.1–6.3, 10.1, and later, S is a surface obtained by blowing up points and contracting a (-2)- or (-3)-curve, so S has a quotient singularity of type 1/2(1,1) or 1/3(1,2) on the exceptional divisor E. The text asserts the resulting intersection formulas and the different Δ_E = (1/2)P + λQ (or 2/3 P + λQ) but does not check that these models are plt blowups of the pair or that the different is klt. If any of these singular models fails the hypotheses of [5, Thm 1.7.9], the equality in Theorem 6.4 and in the quartic cases of the Main Theorem is unsupported. This is load-bearing because the smooth-quartic formulas, the A-singularity formulas, and the K-stability corollaries all inherit the unverified lower bound.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper computes the δ-invariant of log Fano pairs (P2, λC_d), where C_d is a plane curve of degree d ≤ 4. After setting up Fujita's A/S formalism and the Abban–Zhuang lower-bound inequality, the author proves a sequence of local lemmas for each singularity type or tangency configuration, then assembles them into a Main Theorem giving exact formulas on prescribed λ-intervals. The paper also derives K-stability corollaries for double covers and complete intersections, and includes classification tables for cubic and quartic curves.","tokens_in":57972,"tokens_out":9767,"duration_ms":107031,"significance":"If the computations are correct, this is a valuable reference result: it gives the first complete list of exact δ-invariants for log Fano planes with boundary of degree up to 4, with formulas depending only on the worst singularity and on the existence of high-order tangent lines. The method is genuinely from first principles—no fitted parameters appear, and the formulas have sharp upper and lower bounds. The paper also shows how these surface values can feed into Abban–Zhuang reductions for threefolds. My main reservation is that the lower-bound half of each equality rests on a theorem whose hypotheses are not verified in the manuscript; this is a fixable but essential gap.","major_comments":[{"comment":"The equality in the Main Theorem requires, for every weighted blowup model, that π is a plt blowup, E is smooth rational, (E, Δ_E) is klt, and −(K_E + Δ_E) is ample. In Lemmas 4.1, 5.1, 5.2, 6.1–6.3, 8.1, 10.1, 12.1, 13.1, 17.1 and 18.1, the surface S is obtained by contracting (−2)- or (−3)-curves, and E is asserted to have a quotient singularity of type 1/2(1,1), 1/3(1,2), or 1/4(1,3). The text states the resulting intersection form and the formula for the different, but it never verifies that the model is a plt blowup of (P2, λC), that the different is klt, or that the relevant anti-canonical class is ample. Since (2.0.2) is the only source of the lower bound, the claimed equalities are not proved as written. Please add a case-by-case verification, or a general lemma covering these singular models.","section":"§2, Eq. (2.0.2)"},{"comment":"The local lemmas compute δ_P only for points P ∈ C. The global δ is defined as an infimum over all P ∈ P2, and the paper does not bound δ_P for P ∉ C, nor does it systematically check all smooth points of every component in the reducible cases. For instance, Lemma 4.1 proves δ_P = 1 only at points P on a smooth conic; the case P ∉ C is not treated, and Theorem 7.4 similarly jumps from a local statement at an A1 point to a global equality without checking the remaining points. The missing cases may be harmless—a simple blowup computation shows δ_P ≥ 1 off the curve—but they must be stated explicitly for the exact global claims to follow.","section":"Main Theorem / Section 7"},{"comment":"The statement of Lemma 6.3 is inconsistent with its own construction. The lemma says the tangent line L has multiplicity 3, but the construction uses σ*(C) = C + 4E, A(P2,λC)(E) = 5 − 4λ, and E^2 = −1/4, which is the multiplicity-4 (hyperflex or 4-tangent) case. The second smooth-quartic formula in the Main Theorem, δ = (3/5)(5−4λ)/(3−4λ), is exactly the multiplicity-4 case, so the lemma should be restated accordingly, and the proof should be checked against the correct statement.","section":"Lemma 6.3 and Main Theorem, smooth quartic case"}],"minor_comments":[{"comment":"Theorem 8.2 says the cubic curve has an A3 singularity, but the formula given is the A2 formula from Lemma 8.1; the Main Theorem also lists this formula under A2. Please correct the label.","section":"Theorem 8.2"},{"comment":"The first smooth-cubic case, 'no 3-tangent to C3', is vacuous over C: every smooth cubic has flex lines meeting with multiplicity 3. Please either delete this case or specify a different ground field if one is intended.","section":"Main Theorem, smooth cubic case"},{"comment":"The notation in the blowup sequences is often inconsistent, e.g. in Lemma 6.3 the fourth blowup is described as blowing up E2_2 ∩ L2 after E2_2 has already been blown up, and in Lemma 12.1 the indices S4/S5 and E4_4 are scrambled. These errors make the constructions very hard to reproduce.","section":"Various blowup constructions"},{"comment":"Several references are missing or incomplete: [ ?] appears in Remarks 1.13 and 1.15, and [5] is cited as a large book without page or theorem numbers. Please provide complete references and, in particular, the precise statement and hypotheses of the theorem used for (2.0.2).","section":"References"},{"comment":"The tables in Appendix B list formulas for many quartics without specifying the λ-intervals; please make them consistent with the Main Theorem or add a note that the intervals are the same as in the corresponding theorems.","section":"Appendix B"},{"comment":"The application lemmas 1.5 and 1.17 contain several displayed expressions that appear to mix normalization constants (for example, factors involving 3 − λs and 4 − λs are converted differently in successive lines); these lemmas are not used in the proof of the Main Theorem, but they should be corrected or clearly marked as outside the main scope.","section":"Section 1.2"}],"recommendation":"major_revision","confidential_remarks":"To the editor: This is a computational preprint with a large number of small errors, but the central method is recognizable and the results appear plausible. The main correctness gap is the missing verification of the Abban–Zhuang hypotheses for the singular weighted blowup models; this is load-bearing for every equality in the Main Theorem and should be the primary focus of the revision. If the author cannot supply that verification, the exactness claims should be downgraded to upper bounds. The paper would also benefit from an independent check of at least the smooth quartic and A4/A5 formulas."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe paper computes explicit δ-invariants for log Fano pairs (P^2, λC_d) with d ≤ 4, covering smooth, singular, and non-reduced plane curves. That is genuinely useful: the formulas depend on the worst singularity type, and having them collected with interval ranges is a real service. The Abban–Zhuang/Fujita machinery is standard, but the systematic classification and the explicit computations are new, even if individual cases were known. I expect these formulas to be used in higher-dimensional K-stability reductions.\n\nThe good stuff: the main theorem covers reduced and non-reduced curves, including A1–A7, D4–D6, E6–E7, multiple lines, double conics, etc. The appendices with normal forms and δ-values at λ = 1/2 are helpful. The applications to quartic double solids, sextic double solids, and cubic threefolds are quick corollaries but do show the formulas have teeth.\n\nThe soft spots are real but mostly cosmetic. There are numerous typos that affect statements: Lemma 6.3 says “multiplicity 3” while the construction and formula are for a 4-tangent; Theorem 8.2 labels the A2 case as A3; the “no 3-tangent” case for smooth cubics over C is vacuous since every smooth cubic has flex points. These are fixable, but a referee will need to check every lemma.\n\nThe more substantive issue is the reliance on [5, Thm 1.7.9] for the lower bound. The models involve surfaces with quotient singularities on the exceptional divisor E, and the hypotheses (plt blowup, klt different, ampleness) are asserted but never verified case by case. This is not a fatal flaw: these are standard weighted-blowup models and the conditions almost certainly hold, but a serious referee should ask for a proof or a reference covering these specific configurations. The paper would be much stronger if the author added a short section proving a general lemma: for a smooth point P on a plane curve, the blowup-and-contract extraction described in each lemma is a plt blowup with the stated different. That would close the gap.\n\nWho is this for? People working on K-stability of Fano threefolds and on log Fano pairs who need δ-invariants of plane pairs as building blocks. It deserves a serious referee; the formulas are plausible and the classification is valuable, but the text needs careful revision before it can be trusted as a reference. I'd send it to a referee expert in Abban–Zhuang computations, with a specific request to check the hypotheses and the typos.","headline":"Useful systematic formulas for δ-invariants of log Fano plane pairs, but the lower-bound hypotheses need verification and the typos are numerous.","tokens_in":58473,"tokens_out":4271,"would_cite":true,"duration_ms":48584,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14J45","14J26","14H50"],"pacs":[],"model":"deepseek-v4-flash","headline":"Exact stability numbers are pinned down for plane curves of degree up to 4, with the $\\delta$-invariant given by rational functions of the coefficient $\\lambda$.","keywords":["delta-invariant","K-stability","log Fano pairs","plane curves","ADE singularities","Zariski decomposition","log canonical threshold"],"falsifier":"For the explicit quartic $x(x^2z+xz^2+\\beta xyz+y^3)=0$ with an $A_5$ singularity whose tangent line is the component $x=0$, the paper's table predicts $\\delta(\\mathbb{P}^2,\\lambda C)=\\frac{3}{4}\\cdot\\frac{4-6\\lambda}{3-4\\lambda}$; evaluating at $\\lambda=\\frac12$ gives $\\frac34$. Computing the same invariant directly from the definition as the infimum of $A/S$ over all prime divisors on this curve would settle the formula.","tokens_in":57473,"feed_emoji":"📐","tokens_out":8266,"duration_ms":89046,"temperature":0.7,"pith_summary":"The paper computes, exactly, the $\\delta$-invariant of the log Fano pair $(\\mathbb{P}^2, \\lambda C_d)$ where $C_d$ is a plane curve of degree $d \\le 4$ and $\\lambda$ lies in a specified interval. The answer is a piecewise rational function of $\\lambda$ that depends only on the worst singularity type of $C_d$; for smooth quartics it also depends on whether the curve admits a 4-tangent line. Since $\\delta > 1$ is equivalent to K-stability, these formulas decide K-stability and K-semistability for all such pairs. The paper records applications in which stability of higher-dimensional Fano varieties is reduced to these plane-curve computations.","feed_headline":"Exact stability numbers pinned down for plane curves up to degree 4","feed_subtitle":"The δ-invariant equals a rational function of λ fixed by the worst singularity of the curve.","key_machinery":"The mechanism is the local ratio $A/S$: for a prime divisor $E$ over $\\mathbb{P}^2$, $A$ is the log discrepancy of $(\\mathbb{P}^2, \\lambda C_d)$ at $E$ and $S$ is the integral of volumes $\\mathrm{vol}(f^*(-K_{\\mathbb{P}^2} - \\lambda C_d) - vE)$ up to the pseudo-effective threshold. The two lower-bound formulas quoted from the paper's reference [5] reduce $\\delta$ to the minimum of such ratios over exceptional divisors and, inside each exceptional divisor, over curves, which lets the proofs compute $\\delta$ by writing out Zariski decompositions on explicitly constructed blowups. The constructions include weighted blowups whose exceptional curves carry quotient singularities such as $\\frac{1}{2}(1,1)$ and $\\frac{1}{3}(1,2)$, and the boundary divisor on the exceptional curve incorporates the pullback of $\\lambda C_d$.","core_discovery":"On the paper's own terms, the central discovery is that for degrees $d \\le 4$ and the stated $\\lambda$-intervals, the invariant $\\delta(\\mathbb{P}^2, \\lambda C_d)$ is not merely bounded but exactly equal to explicit rational functions. For example, $\\delta(\\mathbb{P}^2, \\lambda C_4) = \\frac{3}{4}\\cdot\\frac{4-3\\lambda}{3-4\\lambda}$ for a smooth quartic without a 4-tangent, $\\delta(\\mathbb{P}^2, \\lambda C_4) = \\frac{3}{5}\\cdot\\frac{5-4\\lambda}{3-4\\lambda}$ when a 4-tangent exists, and similarly determined formulas hold for each singularity type $A_1$ through $A_7$, $D_4$ through $D_6$, $E_6$, $E_7$, and the four-line singularity, as well as for non-reduced curves built from double and triple components. The whole list is organized by the worst singularity of $C_d$, so that knowing the singularities of the curve is enough to write down $\\delta$.","pith_inferences":["A natural next step, not taken in the paper, is to extend the same blowup-chain construction to degree 5 curves; the much larger classification of quintics suggests the list of cases would grow substantially rather than collapse into a few formulas.","The interval cutoffs (for example $\\lambda \\in [\\frac{3}{8}, \\frac{7}{10}]$ for $A_4$) suggest places where the chosen divisor stops being the minimizer; computing $\\delta$ outside those intervals could reveal new transitions and is a direct test of the method.","The dependence on the existence of a 4-tangent for smooth quartics indicates that global tangency configurations, not just local singularities, control the invariant; counting multiple 4-tangents or higher tangencies would be a natural extension.","One could test the formulas numerically on random plane quartics by computing $A/S$ for finitely many divisors; agreement would support the exactness, while finding a smaller ratio would point to missing divisors."],"forward_implications":["Each formula gives a sharp K-stability boundary: within the stated intervals the pair is K-stable when the value exceeds 1 and K-semistable when it equals 1.","The plane-curve formulas plug directly into the reductions in Section 1.2, producing K-stable examples among Du Val del Pezzo surfaces of degree 2, cubic threefolds, quartic and sextic double solids, and complete intersections.","The computations supply new explicit examples of K-stable and K-semistable log Fano pairs, including pairs with singular curve components.","Because the main theorem lists values separately for each worst singularity type, it can be used as a lookup table for $\\delta(\\mathbb{P}^2, \\lambda C_d)$ whenever a plane curve of degree at most 4 is presented."],"supporting_citations":[{"why":"Supplies the two lower-bound formulas used to turn $\\delta$ into a minimum over exceptional divisors and their curves.","marker":"[5]"},{"why":"Provides the log canonical thresholds of plane curves that fix the admissible $\\lambda$-intervals.","marker":"[8]"},{"why":"Shows K-stability of finite covers reduces to stability of the branch plane pair, used in the applications.","marker":"[16]"},{"why":"Classification of plane quartic curves used to enumerate the singularity types covered by the main theorem.","marker":"[18]"},{"why":"Defines the $\\delta$-invariant and the criterion $\\delta>1$ characterizes K-stability, the central link used here.","marker":"[19]"}],"fun_headline_variants":["Exact δ-invariants for log Fano planes, degree ≤ 4","δ-invariant exact formulas for plane curves d ≤ 4","New K-stable examples from exact δ computations","Worst singularity determines δ for plane curves d ≤ 4","Exact δ formulas for log Fano pairs from plane curves"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The paper assumes the two lower-bound formulas from reference [5] remain valid for every intermediate surface used in the proofs, including those with exceptional divisors having quotient singularities, although it does not verify the technical hypotheses case by case.","fun_headline_variants_meta":{"raw":{"variants":["Exact δ-invariants for log Fano planes, degree ≤ 4","δ-invariant exact formulas for plane curves d ≤ 4","New K-stable examples from exact δ computations","Worst singularity determines δ for plane curves d ≤ 4","Exact δ formulas for log Fano pairs from plane curves"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000671,"raw_usage":{"total_tokens":3011,"prompt_tokens":850,"completion_tokens":2161,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":466,"completion_tokens_details":{"reasoning_tokens":2088}},"tokens_in":466,"tokens_out":2161,"duration_ms":17407,"temperature":1.0,"reasoning_tokens":2088,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T13:06:09.551245+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For the explicit quartic $x(x^2z+xz^2+\\beta xyz+y^3)=0$ with an $A_5$ singularity whose tangent line is the component $x=0$, the paper's table predicts $\\delta(\\mathbb{P}^2,\\lambda C)=\\frac{3}{4}\\cdot\\frac{4-6\\lambda}{3-4\\lambda}$; evaluating at $\\lambda=\\frac12$ gives $\\frac34$. Computing the same invariant directly from the definition as the infimum of $A/S$ over all prime divisors on this curve would settle the formula.","supporting_citations":[{"cited_title":"Araujo, A.-M","cited_arxiv_id":null,"evidence_quote":"Supplies the two lower-bound formulas used to turn $\\delta$ into a minimum over exceptional divisors and their curves."},{"cited_title":"Cheltsov, Worst singularities of plane curves of given degree Journal of Geometric Analysis 27 (2017), 2302-2338","cited_arxiv_id":null,"evidence_quote":"Provides the log canonical thresholds of plane curves that fix the admissible $\\lambda$-intervals."},{"cited_title":"Dervan,On K-stability of finite covers , Bull","cited_arxiv_id":null,"evidence_quote":"Shows K-stability of finite covers reduces to stability of the branch plane pair, used in the applications."},{"cited_title":"HuiPlane Quartic Curves, Thesis Ph.D., Liverpool University, (1979)","cited_arxiv_id":null,"evidence_quote":"Classification of plane quartic curves used to enumerate the singularity types covered by the main theorem."},{"cited_title":"Fujita, A valuative criterion for uniform K-stability of Q-Fano varieties , J","cited_arxiv_id":null,"evidence_quote":"Defines the $\\delta$-invariant and the criterion $\\delta>1$ characterizes K-stability, the central link used here."}],"review_version":1}