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Rigid affine cones over singular del Pezzo surfaces

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Pith's one-line read The paper completely classifies which affine cones over index-one log del Pezzo surfaces admit a nontrivial \( \mathbb{G}_a \) action, showing that the only case with an anticanonical polar cylinder is the complete intersection of two…

desk verdict The paper nearly completes the classification, but the final no-cylinder argument for the infinite family S_{2n,2n}, n≥2, does not follow from Proposition 4.8 as written. read the letter →

arxiv 2506.01310 v1 pith:JRSUAQLR submitted 2025-06-02 math.AG

In-Kyun Kim , Jaehyun Kim , Joonyeong Won This is my paper
classification math.AG MSC 14R2014R25
keywords G_a-actionanticanonicalpolarcylinderweightedprojectivespacelogdelPezzosurfaceaffineconerigiditycanonicalthreshold
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 asks which affine cones over mildly singular del Pezzo surfaces admit a nontrivial additive-group (\( \mathbb{G}_a \)) action, and answers the question completely for the index-one surfaces that live in weighted projective spaces. The answer is sparse: a cone is rigid except when the surface is the complete intersection of two quadric hypersurfaces in \( \mathbb{P}^4 \). Equivalently, that single surface admits an anticanonical polar cylinder, and no other quasi-smooth well-formed complete intersection log del Pezzo surface of index one does. The proof combines a threshold criterion that excludes most cases with a detailed study of four exceptional surfaces, including the infinite family \( S_{2n,2n} \subset \mathbb{P}(1,1,n,n,2n-1) \).

What carries the argument

The load-bearing tool is the log canonical threshold \( \operatorname{lct}(S,-K_S) \) of the anticanonical divisor. For a Fano variety with klt singularities, \( \operatorname{lct}(S,-K_S) \ge 1 \) excludes the existence of an anticanonical polar cylinder, by Theorem 3.1. The paper combines this criterion with known threshold tables that cover most hypersurfaces and complete intersections, leaving four surfaces where thresholds do not settle the question. For those four, the argument uses local non-log-canonicity estimates together with the structure of the pencil induced by a cylinder, and for the final family it uses repeated contractions of \( (-1) \)-curves to reduce to a weighted projective plane.

What would settle it

Recompute the log canonical threshold for one of the complete intersection surfaces listed in Table 2 with a stated lower bound at least one, such as No. 10 with bound \( 9/8 \); if the true threshold were below one, Theorem 3.1 would no longer exclude a cylinder, and the classification would be incomplete. A direct refutation would be to exhibit an explicit anticanonical polar cylinder on \( S_{2n,2n} \) for some \( n \ge 2 \), contradicting Theorem 4.9.

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Extended reading notes

Core claim

The central claim is the Main Theorem: for any quasi-smooth well-formed complete intersection log del Pezzo surface \( X_{d_1,\dots,d_{n-2}} \) in a weighted projective space \( \mathbb{P}(a_0,\dots,a_n) \) of index one, an anticanonical polar cylinder exists if and only if the surface is the complete intersection of two quadrics in \( \mathbb{P}^4 \). Since such a cylinder is equivalent to a nontrivial \( \mathbb{G}_a \) action on the affine cone by the cited dictionary, the paper concludes that all other affine cones over these surfaces are rigid. The proof splits the classification into surfaces excluded by the log canonical threshold being at least one and four exceptional surfaces handled by local non-log-canonicity estimates and an explicit birational reduction. For the family \( S_{2n,2n} \subset \mathbb{P}(1,1,n,n,2n-1) \), it constructs a birational morphism to \( \mathbb{P}(1,1,2n-1) \) and shows that a cylinder exists exactly for \( n=1 \).

Load-bearing premise

The proof depends on the imported log canonical threshold values and lower bounds for most surfaces in Tables 1 and 2; if any of the thresholds claimed to be at least one were actually smaller than one, a cylinder could exist among the surfaces the paper excludes by Theorem 3.1.

Editorial extensions

If this is right

  • For every quasi-smooth well-formed complete intersection log del Pezzo surface of index one except the two-quadric intersection in \( \mathbb{P}^4 \), the affine cone is rigid and admits no nontrivial \( \mathbb{G}_a \) action.
  • The single exceptional cone, over the smooth del Pezzo surface of degree four, carries a nontrivial \( \mathbb{G}_a \) action realized through an anticanonical polar cylinder.
  • The infinite family \( S_{2n,2n} \subset \mathbb{P}(1,1,n,n,2n-1) \) is completely settled: it contains a cylinder precisely when \( n=1 \), even though for \( n \ge 2 \) its log canonical threshold is below one, so threshold methods alone cannot decide this case.
  • The answer to Question 1.3 is affirmative but extremely sparse: among all index-one cases, exactly one surface answers it.
  • The birational morphism constructed in Section 4 shows that intermediate blowups of \( \mathbb{P}(1,1,2n-1) \) admit ample polar cylinders, although the terminal surface \( S_{2n,2n} \) does not for \( n \ge 2 \), as noted in Remark 4.10.

Reading between the lines

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

  • We infer that the threshold criterion \( \operatorname{lct} \ge 1 \) is likely the sharp exclusion tool for nearby families as well, provided the corresponding threshold values can be computed.
  • The contraction argument for \( S_{2n,2n} \) suggests a concrete recipe for writing down explicit \( \mathbb{A}^1 \)-fibrations on the intermediate surfaces and for testing whether rigidity persists under small deformations of the defining equations.
  • One could ask whether the \( \mathbb{G}_a \) action on the cone over the smooth quartic del Pezzo is the only algebraic action on that cone, extending the classification from additive actions to full automorphism groups.
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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 quasi-smooth well-formed complete intersection log del Pezzo surfaces of index one in weighted projective spaces. It claims that such a surface admits an anticanonical polar cylinder if and only if it is a complete intersection of two quadrics in P^4. The proof has three parts: a reduction using log canonical threshold computations from earlier work that excludes most surfaces, a direct local argument excluding three exceptional surfaces (S_10, S_15, S_6,8), and a birational analysis of the family S_{2n,2n} in P(1,1,n,n,2n-1). The paper concludes that affine cones over all these surfaces are rigid except in the smooth quartic del Pezzo case.

Significance. If the main theorem is correct, it completes the classification of anticanonical polar cylindricity for this class of surfaces and identifies exactly one non-rigid family among them. The paper contains explicit tables of surfaces and log canonical thresholds, and the positive result for n=1 is consistent with the known smooth case. The detailed intersection-number computations in Lemma 4.2 and Lemma 4.6 are useful and appear mostly self-contained. However, two load-bearing points need attention: the final contradiction for S_{2n,2n} with n≥2 is logically incomplete as written, and the main theorem is stated more broadly than the classification used in the proof. The negative part for the vast majority of surfaces also rests on lct values imported from [15,16]; this is legitimate delegation to prior work, but it means the completeness of the classification is conditional on those computations.

major comments (3)
  1. [Section 4, Theorem 4.9 and Proposition 4.8] The final contradiction in Theorem 4.9 does not follow from Proposition 4.8. Proposition 4.8 asserts only that for each non-lc pair (S,D) there exists some effective anticanonical divisor T with Supp(T)⊂Supp(D). In its proof, T is constructed from the center q∈E of the non-lc valuation after the weighted blow-up at p_w, and different non-lc pairs can have different centers q and hence different divisors T. After Lemmas 2.4 and 2.5 produce D' with Supp(T)⊄Supp(D'), applying Proposition 4.8 to the pair (S,D') gives a possibly different divisor T' with Supp(T')⊂Supp(D'), which is not a contradiction. To close the argument one would need to prove that the center q, or equivalently the divisor T, is the same for D and D', or that every effective anticanonical divisor supported in any non-lc boundary is non-lc at p_w. Neither is shown. Thus the absence of cylinders for S_{2n,2n} with n≥2 is not established by the present proof.
  2. [Section 4, Proposition 4.8] In the displayed alternatives at the end of the proof of Proposition 4.8, the divisor that is Q-linearly equivalent to -K_{Y0} is eT+2L_i or eT+L_i, not eT alone. Since L_i is not contracted by π0, the image of eT under π0 is generally not an anticanonical divisor on S. The proof should state that the pushed-forward anticanonical divisor is the image of eT+mL_i; because the earlier argument shows L_i⊂Supp(eD) when L_i passes through q, the support of that divisor is still contained in the boundary. As written, the class computation is inaccurate.
  3. [Section 1, Main Theorem and Corollary 1.4; Section 3, Corollary 3.3] The main theorem is stated for an arbitrary complete intersection surface X_{d_1,...,d_{n-2}} in P(a_0,...,a_n), but the proof and the cited Theorem 3.2 from [15,16] only cover hypersurfaces and codimension-two complete intersections in P(a_0,...,a_4). The paper does not explain why quasi-smooth well-formed complete intersection log del Pezzo surfaces of index one in higher codimension can be excluded or reduced to these cases. Unless such a reduction is supplied or cited, the theorem should be restricted to the codimension-two setting that the proof actually treats.
minor comments (4)
  1. [Abstract and Introduction] The abstract says the paper completely determines anticanonical polar cylinders in 'quasi-smooth log del Pezzo surfaces of index one,' but the theorem and proof concern complete intersection log del Pezzo surfaces in weighted projective spaces. The scope should be stated precisely to avoid overclaiming.
  2. [Section 2, Lemma 2.4] The notation D_i is introduced without saying that the D_i are the distinct irreducible components of Supp(D). This makes the condition T=∑ b_i D_i ambiguous; please clarify.
  3. [Section 3, diagram (3.7)] The diagram has unlabelled arrows and the phrase 'resolution of indeterminacy \tilde\rho: W→P^1 at p' is imprecise; one resolves the indeterminacy of the rational map ρ, and the base point p should be described as the center of the resolution.
  4. [Section 5, Tables 1 and 2] The tables combine exact lct values with lower bounds. The authors should state explicitly which entries are exact and which are only lower bounds, and indicate that the tables are quoted from [2,3,5,13,15,16] rather than recomputed here.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular reduction found: imported lct and uniqueness results are parameter-free external theorems, and the flagged Theorem 4.9 gap is a validity concern, not a circularity.

full rationale

Walking the derivation chain, the only-if direction for the bulk of the classification delegates lct >= 1 to [15,16] and then applies Theorem 3.1 to exclude cylinders. These are same-group citations, but they state parameter-free results whose assumptions do not include cylindricity or the Main Theorem; under the review rules they count as independent support and do not constitute circularity. The four exceptional surfaces are handled by in-paper arguments: Lemmas 3.5-3.6 and Theorem 3.9 for S10, S15, S6,8, and Section 4 for S_{2n,2n}. Proposition 4.8 is proved inside the paper, using a uniqueness statement from [6]; that, too, is a parameter-free external theorem, not an assumption of the target claim. The skeptic's concern about Theorem 4.9 is real but is not circularity: Proposition 4.8 gives, for each non-lc boundary D, some anticanonical T supported in D, while the contradiction step needs a single T common to all non-lc boundaries. That is a quantifier-shift gap for correctness review, not a reduction of the conclusion to its input by construction. No fitted parameter is renamed as a prediction, and no equation is equivalent to its input by definition. Therefore the circularity score is 0.

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

The central claim rests on the completeness of the weighted projective classification, the correctness of prior lct computations, and a known theorem linking lct to cylinder absence. No free parameters or invented entities are introduced.

assumptions (5)
  • standard math Base field is algebraically closed of characteristic zero
    Stated in the opening paragraph; used throughout for the algebraic geometry constructions.
  • domain assumption The lists in Tables 1 and 2 exhaust all quasi-smooth well-formed complete intersection log del Pezzo surfaces of index one in weighted projective spaces
    The paper relies on the classifications of [24] and [15]; if a surface is missing, the 'if and only if' could fail. Section 5 and the Main Theorem.
  • domain assumption The log canonical threshold values and lower bounds in Tables 1 and 2 are correct
    Theorem 3.2 from [15,16] supplies lct >= 1 for all but four surfaces, and the absence proof depends on it. The values are not derived in this paper.
  • domain assumption Theorem 3.1 from [4]: a Fano variety with lct at least one has no anticanonical polar cylinder
    Used to exclude cylinders for all non-exceptional surfaces in Corollary 3.3.
  • domain assumption The existence of an anticanonical polar cylinder on the complete intersection of two quadrics in P4 is known from [17]
    The 'if' direction of the Main Theorem for n=1 is cited rather than proved.

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Pith. "Pith review of Rigid affine cones over singular del Pezzo surfaces." pith.science (2026). https://pith.science/paper/JRSUAQLR

@misc{pith2026250601310,
  author       = {Pith},
  title        = {Pith review of: Rigid affine cones over singular del Pezzo surfaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JRSUAQLR}},
  note         = {Machine review of arXiv:2506.01310}
}
read the original abstract

We completely determine the existence of anticanonical polar cylinders in quasi-smooth log del Pezzo surfaces of index one.

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Cited by 1 Pith paper

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

  1. Cylindricity of weighted singular del Pezzo surfaces over fields of characteristic zero

    math.AG 2025-12 conditional novelty 6.0 of 10

    For k-forms of blow-ups of P(1,1,m), rationality and cylindricity are determined, in the covered cases, by the blow-up count n, the Galois-orbit invariant ℓ_S, and the k-points of the exceptional curve Q.

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