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K-stability of del Pezzo surfaces with a single quotient singularity

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2507.13649 v1 pith:JP27KXUU submitted 2025-07-18 math.AG

In-Kyun Kim , Dae-Won Lee This is my paper · ORCID
classification math.AG MSC 14J4514J1714E3014J50
keywords K-stabilitydelPezzosurfacesquotientsingularitydelta-invariantAbban-ZhuangtheoryKähler-EinsteinmetricsZariskidecompositionFanovarieties
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 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.

What carries the argument

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 $\delta > 1$.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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 $n,m$.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

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.

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 (3)
  1. [§3.2.2, Lemma 3.9] 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.
  2. [§3.2.2, Lemma 3.9] 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.
  3. [§3.2.5, Lemma 3.20] 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.
minor comments (4)
  1. [§3.2.2, Lemma 3.8] 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'.
  2. [§3.2.2, Lemma 3.9] 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.
  3. [§3.2.1, Lemma 3.5] 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.
  4. [§3.2.5, Lemma 3.20] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation uses external δ-invariant criteria and direct computations, with no fitted parameters renamed as predictions.

full rationale

The paper's derivation is self-contained against external benchmarks. The K-unstable range n+m≥8 is proved by applying Liu's volume bound (Theorem 2.17) to the anticanonical degree formulas computed directly from the resolution data, not from the target classification. The K-stable cases are proved by lower bounds on the δ-invariant via the Abban–Zhuang theory; the constant λ=3/4 is justified by the explicit log-canonicity statement of Lemma 3.5 and by the inequalities in Theorem 2.16, so it is a proven bound rather than an assumed parameter. The cases (2,2) are cited from the independent δ-invariant computations of Denisova and of Odaka–Spotti–Sun. The only fragile step is Lemma 3.9, where the identification of the weighted blow-up with weights (1,4) at the singular point of the surface in Figure 4 with the surface in Figure 5, together with the discrepancy formula KS'=f*(KS)+(2/3)E and E2=-3/4, is asserted without proof or reference. That is a genuine correctness risk for the smooth-point bound of S7_4,2 and for the later transfer arguments, but it is not circular: the identification is a geometric statement about the surfaces, and it does not presuppose K-stability of S7_4,2 nor define any invariant in terms of the conclusion. No fitted parameters are relabeled as predictions, no load-bearing self-citation chain is used, and no uniqueness theorem from the authors' prior work is imported to force the classification.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters were fitted; the paper proves inequalities rather than tuning constants. The computations rely on established theorems in K-stability and on the explicit geometry of the constructed surfaces, with the main unproved burden being the weighted blow-up identification and the general position assumptions.

assumptions (5)
  • standard math The delta-invariant criterion: delta(X) > 1 iff uniformly K-stable, delta(X) = 1 iff K-semistable (Blum-Jonsson, Theorem 2.14)
    Used to convert delta computations into K-stability statements in Theorem 1.4 and throughout Section 3.
  • standard math Abban-Zhuang lower bound for local delta-invariant via admissible flags (Theorem 2.20)
    The main computational tool; every delta_p bound relies on this inequality.
  • standard math Liu's volume bound: for a quotient singularity with local group G, K-semistability implies (-K_X)^n <= (n+1)^n/|G| (Theorem 2.17)
    Used in Theorem 3.2 to rule out K-stability for n+m >= 8 cases.
  • domain assumption General position of the blown-up points in Definitions 1.2 and the constructions of Section 3
    The intersection numbers and the identification S^n_{n,m} congruent to S^m_{m,n} rely on points being chosen generally.
  • ad hoc to paper The weighted blow-up f: S' -> S in Lemma 3.9 is a birational morphism to the surface of Figure 4
    Asserted without proof; used to apply Lemma 3.5 to the S^7_{4,2} smooth-point argument.

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Pith. "Pith review of K-stability of del Pezzo surfaces with a single quotient singularity." pith.science (2026). https://pith.science/paper/JP27KXUU

@misc{pith2026250713649,
  author       = {Pith},
  title        = {Pith review of: K-stability of del Pezzo surfaces with a single quotient singularity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JP27KXUU}},
  note         = {Machine review of arXiv:2507.13649}
}
abstract

In this paper, we study the K-stability of del Pezzo surfaces with a single quotient singularity whose minimal resolution admits exactly two exceptional curves \(E_1\) and \(E_2\) with \(E_{1}^2=-n\), \(E_{2}^2=-m\) for \(n,m\geq 2\).

Figures

Figures reproduced from arXiv: 2507.13649 by the authors.

Figure 1
Figure 1. del Pezzo surfaces with a single singularity of type 1 n (1, 1) Note that S n+1 n can be embedded as a degree n + 1 hypersurface in P(1, 1, 1, n), and can also be obtained by blowing-up P 2 at n + 1 points on a line L and then contracting the strict transform of L. See Example 2.6 and Proposition 2.8. Theorem 1.1 (cf. [23, Theorem B]). Let n ≥ 4 be a positive integer, and S a del Pezzo surface which is isomorphic to… view at source ↗
Figure 2
Figure 2. Construction of S 0 n,m Let S k n,m be the surface obtained by blowing-up at k distinct smooth general points on S 0 n,m. We note that the surface S n n,m can also be obtained by blowing-up points on the surface P(1, 1, m). Let ℓ2 ∈ |OP(1,1,m) (1)| be a curve, and πm : S˜ 2 → P(1, 1, m) a blow-up at n distinct smooth points on ℓ2. By contracting the strict transform L2 of ℓ2, we obtain the birational morphism φm : S… view at source ↗
Figure 3
Figure 3. del Pezzo surfaces with a single singularity of type 1 mn−1 (1, n) Remark 1.3. Assume that n ≥ m ≥ 2. The surface S n+3 n,m is still a singular del Pezzo surface with a singularity of type 1 mn−1 (1, n) if and only if (n, m) is one of the following: (2, 2),(3, 2) or (4, 2). The surfaces in [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

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