Conical K\"ahler-Einstein metrics on K-unstable del Pezzo surfaces

Dasol Jeong; Jihun Park

REVIEW 3 major objections 5 minor 29 references

Conical K\"ahler-Einstein metrics on K-unstable del Pezzo surfaces

T0 review · 3 major / 5 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read This paper determines the exact optimal cone-angle upper bounds for Kähler–Einstein metrics with conical singularities along smooth anticanonical divisors on the two K-unstable smooth del Pezzo surfaces, namely the blowups of the projective

desk verdict Exact cone-angle bounds for the two K-unstable del Pezzo surfaces, computed via delta-invariants; credible and likely right, but the displayed computations are sloppy enough that the tangent-case lemma needs a careful referee pass. read the letter →

arxiv 2509.08627 v1 pith:J4XGG2MK submitted 2025-09-10 math.AG math.DG

Dasol Jeong , Jihun Park This is my paper · ORCID

classification math.AGmath.DG MSC 14J4514J2632Q2053C25

keywords conicalKähler–EinsteinmetricsdelPezzosurfacesδ-invariantK-stabilityconeangleanticanonicaldivisorMoridreamspacegreatestRiccilowerbound

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper establishes the exact largest cone angle for which a Kähler–Einstein metric with conical singularities along a smooth anticanonical divisor exists on the two smooth del Pezzo surfaces that admit no ordinary Kähler–Einstein metric: the blowup of the projective plane at one point and at two points. On the one-point blowup S1 the threshold is 3/4 if the divisor is tangent to the 0-curve at the intersection with the exceptional curve, and 4/5 otherwise. On the two-point blowup S2 the threshold is 7/9 if the divisor passes through the intersection of two (−1)-curves, and 21/25 otherwise. These values are optimal: they coincide with known upper bounds and show that earlier lower bounds were sharp. The proof computes the δ-invariant of the log pair (Si,(1−λ)Ci) by weighted blowups and a local δ-invariant inequality, then reads off the λ where the invariant first reaches 1.

What carries the argument

The load-bearing object is the δ-invariant of the log Fano pair (Si,(1−λ)Ci), an infimum over divisors of log discrepancy divided by a volume-normalized pseudo-effective threshold; for these pairs a conical metric exists precisely when δ>1. To pin down δ locally, the authors construct weighted (1,m)-blowups at points, contracting part of the exceptional chain to obtain a plt blowup (a mild singularity model whose exceptional divisor is a prime divisor). A local δ-invariant inequality (Theorem 2.4) then bounds the local invariant from below by the minimum of the global A/S ratio and ratios formed from curve-intersection numbers on the exceptional divisor. Each blown-up surface is verified to

What would settle it

Recompute the volume function in Lemma 4.2 from the listed intersection numbers: check whether the positive part (5λ−t)(Ê+2F̂) with negative part (t−4λ)F̂+(t−3λ)L̂ on [4λ,5λ] gives the stated volume, and verify that no extra extremal ray appears in Cone{[Ê],[F̂],[Ĝ],[L̂]}. An error in either would shift the ratio A/S=(4+8λ)/11λ and move the threshold away from 10/13.

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Extended reading notes

Core claim

The central claim is that the conical Kähler–Einstein threshold—the supremum of cone-angle parameters λ for which the equation Ric(ω)=λω+(1−λ)[Ci] has a solution—is exactly the rational value stated in the Main Theorem for every smooth anticanonical divisor on S1 and S2. On S1 the threshold is 3/4 when C1 is tangent to the 0-curve at E∩C1 (equivalently, the blown-up point is an inflection point of the plane cubic), and 4/5 otherwise. On S2 the threshold is 7/9 when C2 passes through the intersection of two (−1)-curves, and 21/25 otherwise. These numbers match the previously established upper bounds, so the earlier lower-bound results are sharp. Because the thresholds are strictly smaller tha

Load-bearing premise

Everything hangs on the asserted generators of the Mori cone of each blown-up surface and the Zariski decompositions computed from them; if any cone description or volume integral is wrong, the A/S ratios and the location of δ=1 change.

Editorial extensions

If this is right

For every smooth anticanonical divisor on S1 and S2, conical Kähler–Einstein metrics exist for all cone angles strictly below the stated threshold and cease to exist at or above it.
The exact values confirm that the conical threshold equals the previously established upper bound in every divisor configuration, so no gap remains between upper and lower estimates.
The strict inequality between the conical threshold and the greatest Ricci lower bound gives a clean family of counterexamples to the 2012 conjecture that the two invariants coincide.
The boundary case λ0 has δ=1, so the pair is strictly K-semistable despite having finite automorphism group; the metric family degenerates exactly at the cone-angle endpoint.
The piecewise-linear formulas for δ over λ∈(0,1] show precisely which divisor governs the invariant in each range.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same weighted-blowup localization is likely to compute conical thresholds for other log del Pezzo surfaces with explicit Mori cones, where currently only bounds are known.
The sharp dependence on tangency and inflection configurations suggests a general principle: the cone-angle threshold is determined by the worst local model where the divisor is tangent to a special curve through the blown-up point, not by the global geometry alone.
Testing the method on higher-dimensional Fano manifolds obtained by blowing up points along anticanonical divisors could reveal whether the thresholds there also arise from simple rational A/S ratios; the surface values provide concrete expected answers to check.
Because the endpoint has δ=1 and finite automorphism group, these examples are natural candidates for studying uniqueness and algebraic degenerations of conical Kähler–Einstein metrics at the maximal cone angle.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The real result here is the exact values R(S1,C1) and R(S2,C2), split by tangency and inflection. Székelyhidi had upper bounds, Cheltsov–Martinez-Garcia had lower bounds, and this paper closes the gap and shows the bound is achieved by the delta-invariant crossing 1. That is a clean, useful within-subfield result, and the paper deserves credit for it.

What is good: the structure is sensible. They reduce to computing local delta-invariants, use Fujita's inequality with weighted blowups, verify the Mori dream space and plt hypotheses via Proposition 2.5, and then assemble piecewise formulas that match all previously known bounds. The final values are consistent: 3/4 versus 4/5 on S1, 7/9 versus 21/25 on S2, and the piecewise delta formulas hit exactly 1 at the claimed thresholds. I see no fitted parameters and no circularity; the target values emerge from the computations rather than being assumed.

The soft spots are real but not fatal. The manuscript is riddled with numerical typos in displayed formulas: (4.6) has 48/25 where the computation gives 48/(25λ); (4.10) writes 48/17 where the lemma needs 12/5; Lemma 4.9 has a threshold 23/60 ≤ 20/23 that should be 23/60 ≤ λ ≤ 20/23, and the lower-bound constant 63/28 appears in the statement but the proof uses 63/23 and 42/23; Lemma 4.14 similarly writes 25/63 while the proof gives λ/3 for one term. These are the kind of errors that make independent verification painful, and in a paper where the whole argument is a long sequence of explicit intersections and Zariski decompositions, they reduce confidence even when the conclusion is plausible.

The stress-test note about Lemma 4.6 is the one place I would push hard. There, q_F is declared an A1 singularity and q_E appears with A=1/2, but q_F is missing from the discrepancy table. If A(q_F)=1/2, then A/S is 6/(13λ) and the claimed δ=1 at λ=3/4 fails. If A(q_F) is actually something else, the displayed table should say so. The lemma also mixes 3+6λ and 3+5λ in the same lower bound, and the final piecewise switches to 48/17 in a way that does not follow from the stated terms. This is a load-bearing claim — the R(S1,C1)=3/4 tangent case relies on it — so it cannot be waved off as a typo. I am not convinced the lemma is wrong, but I am convinced the referee needs to redo that computation line by line.

Verdict: this should go to a serious referee. The method is standard, the main theorem is plausible and supported by the surrounding lemmas, and the S2 half looks cleaner. The S1 tangent case, and the typo cleanup in Section 4 generally, should be fixed before publication. I would not cite the exact constants until a corrected version appears.

Referee Report

3 major / 5 minor

Summary. The paper determines the exact values of R(S_i,C_i), the supremum of cone angles for conical Kähler–Einstein metrics along a smooth anticanonical divisor, for the two K-unstable smooth del Pezzo surfaces S_1=Bl_p P^2 and S_2=Bl_{x_1,x_2} P^2. The Main Theorem states that R(S_1,C_1)=3/4 if C_1 is tangent to the 0-curve at E∩C_1 and 4/5 otherwise, and R(S_2,C_2)=7/9 if C_2 passes through the intersection of two (−1)-curves and 21/25 otherwise. The proof uses the δ-invariant characterization of K-stability, computes or bounds local δ-invariants point by point, and applies Fujita's local δ-formula via weighted blowups. The claimed values agree with the upper bounds of Székelyhidi and the lower bounds of Cheltsov–Martinez-Garcia, and the derivation is self-contained given the cited K-stability theorems. However, the manuscript contains a large number of apparent typographical and endpoint errors in the displayed lower-bound computations, one of which affects the only proof of the tangent S_1 case.

Significance. If the results are correct, they give the first exact optimal cone-angle upper bounds for these two K-unstable del Pezzo surfaces, sharpening Donaldson's conjecture and Székelyhidi's counterexamples. The method is largely synthetic and computational, with no free parameters: the thresholds are obtained by solving δ=1. The values are consistent with all previously known bounds, which is a strong plausibility check. The main weakness is not the overall strategy but the reliability of the many displayed computations; several key displays are internally inconsistent as written and must be corrected before the proof can be accepted.

major comments (3)

[§4.1, Lemma 4.6 and Eqs. (4.13)–(4.14)] This lemma is load-bearing for the tangent case R(S_1,C_1)=3/4, and its proof is not internally consistent as written. (i) The first sentence says q_F is an A1 singularity, but the discrepancy table gives A=1/2 at q_E and has no entry for q_F. If q_F really were A1, its contribution would be (1/2)/(13λ/12)=6/(13λ), which at λ=3/4 is 8/13<1, destroying the equality δ=1. If the intended A1 point is q_E, the text and diagram labels must be corrected throughout. (ii) Eq. (4.14) writes (3+5λ)/(10λ), whereas the upper bound and the lemma statement require (3+6λ)/(10λ); with the displayed (3+5λ), the minimum at λ=3/4 is 9/10<1, not 1. (iii) The final constant 48/17 has no source in the computation; the correct limiting constant is 3. (iv) The Zariski decomposition, volume, and h-function are displayed with intervals ending at 6λ, but the correct support is [2λ,8λ]; with 6λ the stated values S(G
[§4.1–§4.2, lower-bound formulas (4.6), (4.8), (4.10), (4.23)] The same pattern of constant/interval errors appears in several other lemmas. In Lemma 4.2, Eq. (4.6) mixes 48/(25λ) and 48/25; the constant 48/25 is the q_C1 contribution and should be explicitly identified as such. In Lemma 4.3, Eq. (4.8) contains 48/17λ in the displayed list, while the final piecewise uses 48/17; the latter is the q_C1 contribution, and the role of the former is unclear. In Lemma 4.4, Eq. (4.10) concludes with 48/17 although the lemma statement and the preceding lower bound use 12/5; the threshold 5/26 is obtained by equating (1+2λ)/(3λ) with 12/5, not with 48/17. In Lemma 4.9, the statement's min contains 63/28, but Eq. (4.23) concludes with 42/23, and the interval '23/60 ≤ 20/23' is malformed. These errors change the announced piecewise formulas and the thresholds, so the proofs of the corresponding lemmas are not currently verifiable.
[§4.1–§4.2, systematic verification of the Mori-cone and intersection computations] Because the paper hinges on explicit intersection numbers, Mori cone generators, and Zariski decompositions, the many numerical typos make it difficult to verify the load-bearing premise that all constructed surfaces are Mori dream spaces with the stated cone descriptions. I am not claiming that the cone descriptions are wrong—they appear plausible and consistent with Proposition 2.5—but the current text does not allow an independent check of, for example, the claim in Lemma 4.6 that NE(hat S_1)=Cone{[hat E],[hat F],[hat G]}. The authors should provide a clean, systematic account of each weighted blowup: pullbacks, intersection matrix, Zariski decomposition with correct t-intervals, volume function, S-values, and A-values, preferably in tabular form, so that the main theorem's lower bounds are actually established.

minor comments (5)

[§4.1, Lemma 4.2, displayed Zariski decomposition] In the second row of P(t), '(6−λ)F' should read '(6λ−t)F'; the volume formula that follows is consistent with the corrected expression.
[§4.2, Lemma 4.9, displayed interval] The piecewise interval '23/60 ≤ 20/23' is missing the variable λ; it should presumably read '23/60 ≤ λ ≤ 20/23'.
[§4.2, Lemma 4.13, piecewise formula] In the final line, '21+42λ/55' is missing the factor λ in the denominator; it should be (21+42λ)/(55λ).
[§2 and §4, notation] The term '0-curve' is used repeatedly before being defined; it should be introduced as a curve with self-intersection 0 on S_1 (a fiber of the ruling). Also, the label q_F in Lemma 4.6 conflicts with the A-table; the authors should unify the notation for the images of the exceptional curves after contraction.
[§4.2, Lemma 4.15] In the pullback list, 'σ_2^*F = hat F + hat M' refers to a curve F that is not otherwise defined in that lemma; it should presumably be σ_2^*B.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the cone-angle bounds are outputs of δ-invariant computations, not inputs; no fitted parameters or self-citations.

full rationale

The derivation chain is self-contained. The paper computes δ(S_i,(1−λ)C_i) as an explicit piecewise function of λ using Fujita's inequality (Theorem 2.4), the δ-criterion (Theorem 2.3), and a sequence of (1,m)-blowups with explicit Zariski decompositions. The claimed optimal cone angles are then obtained by solving δ=1, e.g. (3+6λ)/(10λ)=1 at λ=3/4 and (4+4λ)/(9λ)=1 at λ=4/5 for S_1; similarly (7+7λ)/(16λ)=1 at λ=7/9 and (3+9λ)/(10λ)=1 at λ=21/25 for S_2. The upper bounds of Székelyhidi and lower bounds of Cheltsov–Martinez-Garcia are cited as context and for comparison, not as inputs to the δ computations. No parameter is fitted to the target R-values; no normalization is chosen to force the answer; and no load-bearing self-citation occurs (all main external theorems are from other authors). The reader's note about a possible arithmetic inconsistency in Lemma 4.6's displayed lower bound (4.14) would be a correctness issue if real, not a circularity one, because the lower bound is derived from the same Zariski-decomposition data rather than assumed from the conclusion.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No numerically fitted parameters appear. The central claim rests on the cited K-stability theorems and explicit Mori cone descriptions; a few geometric counting facts are asserted without proof. No new entities are postulated.

assumptions (7)

domain assumption The Yau-Tian-Donaldson conjecture for log Fano pairs: (X,Δ) with discrete automorphism admits a KE metric iff it is K-stable (Theorem 2.1).
Cited to [19,20]; used to convert existence of conical KE metrics into K-stability of (X,(1−λ)D) and to derive (2.3).
domain assumption K-stability is equivalent to δ>1 (Theorem 2.3).
Cited to [14,17,6]; used to identify the threshold δ=1 as the boundary of existence.
domain assumption Fujita's inequality for local δ-invariants (Theorem 2.4).
Cited to [15]; the main engine for every lower bound in Section 4.
domain assumption Proposition 2.5: after contracting a chain of (−2)-curves of type A_n on a weak del Pezzo surface, the image is a Mori dream space and its Mori cone is spanned by the images of the remaining extremal rays.
Derived from [21,22]; used to justify the Mori cone descriptions for each weighted blowup model.
domain assumption S_1 and S_2 are the only smooth del Pezzo surfaces that do not admit a KE metric (Section 3, first sentence).
Cited to [26]; frames the problem as the only two cases to check.
domain assumption At least three members of each pencil |N_i| are tangent to C_2, and the plane cubic φ_2(C_2) has at least six inflection points outside φ_2(B) (Section 4.2, before (4.36)).
Asserted without proof; needed to ensure that the local computations in Lemmas 4.8-4.10 cover all possible points of C_2.
standard math If two of the three intersection points x_1, x_2, y of the line B with C_2 are inflection points, then all three are (Section 4.2, case analysis).
Standard property of the group law on a smooth plane cubic.

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Cite this review

Pith. "Pith review of Conical K\"ahler-Einstein metrics on K-unstable del Pezzo surfaces." pith.science (2026). https://pith.science/paper/J4XGG2MK

@misc{pith2026250908627,
  author       = {Pith},
  title        = {Pith review of: Conical K\"ahler-Einstein metrics on K-unstable del Pezzo surfaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/J4XGG2MK}},
  note         = {Machine review of arXiv:2509.08627}
}

read the original abstract

We establish the optimal upper bounds for cone angles of K\"ahler-Einstein metrics with conical singularities along smooth anticanonical divisors on smooth K-unstable del Pezzo surfaces.

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

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Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.

[1] [1]

´Equations du type Monge-Amp` ere sur les vari´ et´ es k¨ ahleriennes compactes.C

Thierry Aubin. ´Equations du type Monge-Amp` ere sur les vari´ et´ es k¨ ahleriennes compactes.C. R. Acad. Sci. Paris S´ er. A-B, 283(3):Aiii, A119–A121, 1976

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2013

[3] [3]

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2016

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Berman, S´ ebastien Boucksom, and Mattias Jonsson

Robert J. Berman, S´ ebastien Boucksom, and Mattias Jonsson. A variational approach to the Yau- Tian-Donaldson conjecture.J. Amer. Math. Soc., 34(3):605–652, 2021

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Hacon, and James McKernan

Caucher Birkar, Paolo Cascini, Christopher D. Hacon, and James McKernan. Existence of minimal models for varieties of log general type.J. Amer. Math. Soc., 23(2):405–468, 2010

2010

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Uniqueness of K-polystable degenerations of Fano varieties.Ann

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Ivan A. Cheltsov, Yanir A. Rubinstein, and Kewei Zhang. Basis log canonical thresholds, local intersection estimates, and asymptotically log del Pezzo surfaces.Selecta Math. (N.S.), 25(2):Paper No. 34, 36, 2019

2019