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$\delta$-invariants of log Fano planes

T0 review · 3 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read 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 $\lambda$.

desk verdict Useful systematic formulas for δ-invariants of log Fano plane pairs, but the lower-bound hypotheses need verification and the typos are numerous. read the letter →

arxiv 2505.22528 v1 pith:GQVICX34 submitted 2025-05-28 math.AG

Elena Denisova This is my paper · ORCID
classification math.AG MSC 14J4514J2614H50
keywords delta-invariantK-stabilitylogFanopairsplanecurvesADEsingularitiesZariskidecompositioncanonicalthreshold
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

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.

What carries the argument

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 $\lambda C_d$.

What would settle it

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.

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Extended reading notes

Core claim

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 $\delta$.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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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 / 6 minor

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.

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 (3)
  1. [§2, Eq. (2.0.2)] 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.
  2. [Main Theorem / Section 7] 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.
  3. [Lemma 6.3 and Main Theorem, smooth quartic case] 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.
minor comments (6)
  1. [Theorem 8.2] 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.
  2. [Main Theorem, smooth cubic case] 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.
  3. [Various blowup constructions] 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.
  4. [References] 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).
  5. [Appendix B] 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.
  6. [Section 1.2] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the δ-invariants are derived from Fujita's definition and the external Abban–Zhuang inequalities, with no fitted parameters and no load-bearing self-citations.

full rationale

The paper computes δ(P²,λC_d) by combining an upper bound from an explicit prime divisor E (via A/S ratios) with a lower bound from Abban–Zhuang's inequalities (2.0.1)/(2.0.2), imported from [5], a multi-author text in which the present author does not appear. The construction of each exceptional divisor E and the different Δ_E is dictated by the singularity type of C_d (e.g., Δ_E = ½P + λQ for a ½(1,1) singularity), not by any parameter fitted to the target value of δ. The final formulas are obtained by computing A(X,λC)(E), S(X,λC)(E), and the A/S ratios on E; the matching of upper and lower bounds is a genuine computation, not an identity-by-construction. No self-citation is load-bearing: the author's previous works ([10], [14], [15]) appear only in the reference list and are not used to justify any main claim, and no uniqueness theorem from the author's own work is invoked. The skeptic's concern about unverified plt/klt/ampleness hypotheses for the singular intermediate surfaces (e.g., in Lemmas 4.1, 5.1, 6.1–6.3, 8.1, 10.1) is a possible correctness or rigor gap in applying [5, Thm 1.7.9], but it is not circular: the hypotheses are conditions of an external theorem, and failing to check them would make the lower bound unsupported, not make the derivation reduce to its own conclusion. There are no fitted inputs renamed as predictions, no ansatz smuggled in via citation, and no renaming of a known result. Hence no circular step is exhibited.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim rests on standard theorems in K-stability and on the correctness of lengthy Zariski-decomposition computations. No free parameters are fit to data. No new entities are postulated.

assumptions (4)
  • standard math Fujita's valuative criterion: δ(X,Δ) > 1 iff (X,Δ) is K-stable.
    Invoked in Section 1.1 to relate δ to K-stability.
  • standard math Abban-Zhuang lower-bound formulas (2.0.1) and (2.0.2).
    Used in every lemma to get a lower bound on δ_P; the hypotheses are stated in Section 2 but not verified case by case.
  • domain assumption Zariski decomposition of the relevant divisors on the constructed surfaces is as stated in each proof.
    The positive and negative parts P(v) and N(v) are asserted in each lemma without external verification; these decompositions drive the volume computations.
  • standard math Classification of plane curves of degree ≤ 4 with ADE singularities.
    Cited to references [18], [24], and [30]; used to enumerate cases in the main theorem and appendices.

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Cite this review

Pith. "Pith review of $\delta$-invariants of log Fano planes." pith.science (2026). https://pith.science/paper/GQVICX34

@misc{pith2026250522528,
  author       = {Pith},
  title        = {Pith review of: $\delta$-invariants of log Fano planes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GQVICX34}},
  note         = {Machine review of arXiv:2505.22528}
}
abstract

We compute the $\delta$-invariant for pairs $(\mathbb{P}^2, \lambda C_d)$, where $C_d$ is a plane curve of degree $d \leq 4$. These computations provide new examples of $K$-stable and $K$-semistable log Fano pairs, and contribute to the study of $K$-stability of log Fano varieties via the Abban-Zhuang method, which reduces higher-dimensional problems to the surface case.

Figures

Figures reproduced from arXiv: 2505.22528 by the authors.

Figure 1
Figure 1. Cubic Curves 1 arXiv:2505.22528v1 [math.AG] 28 May 2025 [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Irreducible Quartic Curves [PITH_FULL_IMAGE:figures/full_fig_p067_2.png] view at source ↗
Figure 3
Figure 3. Line and Irreducible cubic 67 [PITH_FULL_IMAGE:figures/full_fig_p067_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Two Conics [PITH_FULL_IMAGE:figures/full_fig_p068_4.png]
Figure 5
Figure 5. Figure 5: Conic and Two Lines "four-line" singularity [PITH_FULL_IMAGE:figures/full_fig_p068_5.png]
Figure 6
Figure 6. Figure 6: Four Lines Appendix D. Non-Reduced Curves In this appendix we provide the pictures which illustrate non-reduced curves for degree less than four. 2 repeated lines [PITH_FULL_IMAGE:figures/full_fig_p068_6.png]
Figure 8
Figure 8. Figure 8: Non-Reduced Cubic Curves repeated conic conic + repeated chord conic + repeated tangent 4 repeated lines 3 repeated lines + 1 line 2 repeated lines + 2 lines (concurrent) 2 repeated lines + 2 lines (transversal) 2 pairs of 2 repeated lines [PITH_FULL_IMAGE:figures/ful…
Figure 9
Figure 9. Figure 9: Non-Reduced Quartic Curves 68 [PITH_FULL_IMAGE:figures/full_fig_p068_9.png]

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

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

    math.AG 2025-09 conditional novelty 6.0 of 10

    For the two K-unstable del Pezzo surfaces, the optimal cone-angle bounds are R(S_1,C_1)=3/4 or 4/5 and R(S_2,C_2)=7/9 or 21/25, depending on tangency or intersection geometry.

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