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On Fano indices of weighted projective spaces

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Pith's one-line read This paper proves that the Fano index of an n-dimensional well-formed weighted projective space with canonical singularities is at most (s_n−1)(2s_n−3), where s_n is the nth Sylvester number.

desk verdict Sharp new bound for Fano indices of weighted projective spaces, but the theorem as stated is false for n=1 and the proof hinges on an unverified t-shifted lemma. read the letter →

arxiv 2608.03434 v1 pith:FUCBZLGO submitted 2026-08-04 math.AG

Haidong Liu This is my paper · ORCID
classification math.AG MSC 14J4514M2552B2052B1152A40
keywords FanoindexweightedprojectivespacecanonicalsingularitiesSylvestersequenceagefunctionintegralsimplexmajorizationtoricvarieties
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 proves that the Fano index of an n-dimensional well-formed weighted projective space with canonical singularities is at most (s_n−1)(2s_n−3), where s_n is the nth term of the Sylvester sequence (s_1=2, s_i=s_1...s_{i-1}+1). This is exactly the bound conjectured to hold for all Fano varieties of dimension n, and the paper shows it for weighted projective spaces and, as a direct corollary, for Q-factorial toric Fano varieties with Picard number one. The proof works by associating to the space an integral simplex whose normalized volume is the Fano index, then using product-sum inequalities and a majorization argument to force the age function at a specific integer to reach n, contradicting the canonical-singularity criterion. The paper also determines the possible Fano indices of 4-dimensional well-formed weighted projective spaces with canonical singularities: they lie in {m>0 | φ(m) ≤ 984}. If the result is right, the earlier upper bound $2y_n^{2}$ is improved to the sharp value for this class, confirming the conjecture in these cases.

What carries the argument

The driving mechanism is the age function age(k) = Σ {a_i k / q}, the sum of fractional parts of the weights scaled by k/q. A criterion (Proposition 2.4) says the weighted projective space has at worst canonical singularities exactly when age(k) ≠ n for every 2 ≤ k ≤ q−2. The proof assumes q > y_n(2y_n−1) and shows that age(2y_n−1) ≥ n, contradicting the criterion. To get there it uses the integral simplex with one interior lattice point associated to the weights: its normalized volume is q, its barycentric coordinates are β_i = a_i/q, and product-sum inequalities (β_1...β_j ≤ β_{j+1}+...+β_{n+1}) together with a majorization inequality force lower bounds on products of β_i. The Sylvester sequence enters through the sharp bounds β_n ≥ 1/(2y_n) and the majorization vector (1/s_1, ..., 1/s_{n-1}, 1/y_n − t).

What would settle it

Check the cited lemma [2, Lemma 4.5]: take a decreasing positive sequence x_1 ≥ ... ≥ x_m with x_1+...+x_m = 1−t and x_1...x_j ≤ x_{j+1}+...+x_m+t for every j, and test numerically whether x_1+...+x_k ≤ 1/s_1+...+1/s_k still holds for 0 < t < 1/y_m; if a counterexample exists, the proof of Lemma 2.11 collapses. Independently, run the dimension-4 search described in Section 4 (or a dimension-5 search) to see whether any well-formed weighted projective space with canonical singularities has Fano index above y_n(2y_n−1).

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Extended reading notes

Core claim

The central claim is Theorem 1.2: for a well-formed weighted projective space X = P(a_1,...,a_{n+1}) of dimension n with at worst canonical singularities and Fano index q = a_1+...+a_{n+1}, one has q ≤ y_n(2y_n−1), where y_n = s_n−1 and s_n is the n-th Sylvester number. The bound is sharp; the example attaining it is the weighted projective space with weights q/s_1, ..., q/s_{n-2}, y_{n-1}, y_{n-1}−1, where q = y_n(2y_n−1). A corollary transfers the bound to any n-dimensional Q-factorial toric Fano variety with Picard number one and canonical singularities, by pulling back to a well-formed weighted projective space via a finite étale-in-codimension-one morphism. In dimension 4, the proof is supplemented by a computation showing the Fano index lies in the set {m ∈ Z_{>0} | φ(m) ≤ 984}.

Load-bearing premise

The proof depends on a lemma from an earlier paper that is invoked without its full statement; the lemma must hold when the usual product-sum inequality has an extra positive term on the right, and the paper gives no proof of that extension.

Editorial extensions

If this is right

  • The conjectured bound y_n(2y_n−1) is now established for all well-formed weighted projective spaces with canonical singularities, improving the previous general bound 2y_n^2 for this class.
  • The same bound holds for Q-factorial toric Fano varieties with Picard number one and canonical singularities, since each such variety is covered by a well-formed weighted projective space with no larger Fano index.
  • If the paper is right, any further improvement of the Fano-index bound for general Fano varieties must come from phenomena not visible in the toric, Picard-number-one case.
  • In dimension 4, the Fano index of a well-formed weighted projective space with canonical singularities is at most 3486, and it always satisfies φ(q) ≤ 984; this matches the pattern observed in lower dimensions.
  • The conjectured coincidence of index sets (Conjecture 4.2) would imply, if it holds in dimension 4, that terminal Calabi–Yau 4-folds have index at most 3486.

Reading between the lines

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

  • The majorization technique used here may extend to other Fano varieties whose associated simplices satisfy product-sum inequalities, potentially turning the bound y_n(2y_n−1) from a conjecture into a theorem for wider classes.
  • The divisibility predictions in Conjecture 4.8 suggest that for large q, Fano indices of these spaces appear in arithmetic progressions with step y_n; if true, this would give a structural explanation for the gap between consecutive indices.
  • The bound φ(q) ≤ φ(y_n(2y_n−1)) in dimension 4 hints at a general relationship between Fano indices and Euler's totient that could be tested computationally for n=5.
  • If the conjectured coincidence of index sets holds in all dimensions, then the Fano-index bound for weighted projective spaces would also bound the indices of terminal Calabi–Yau varieties, connecting the two classifications.
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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

5 major / 5 minor

Summary. The paper studies the Fano index q of a well-formed weighted projective space P(a_1,...,a_{n+1}) with canonical singularities, where q is the sum of the weights. The main result (Theorem 1.2) claims the bound q ≤ y_n(2y_n−1), with y_n = s_n−1 and s_n the Sylvester sequence term. The proof, given for n≥4, combines the age-function criterion of Kasprzyk with lattice-simplex bounds from Averkov–Krümpelmann–Nill, Pikhurko, and Hensley, plus a majorization argument. The paper also derives a corollary for Q-factorial toric Fano varieties with Picard number one, and in Section 4 it presents a computer-assisted classification of Fano indices in dimension 4 together with several conjectures relating these indices to indices of terminal Calabi–Yau varieties and log canonical singularities.

Significance. If correct, the result sharpens the previously known bound q ≤ 2y_n^2 for this class to the conjecturally sharp value y_n(2y_n−1), thereby confirming a conjecture of Chengxi Wang for weighted projective spaces and fake weighted projective spaces. The proof strategy is interesting: it uses the age-function criterion to convert an arithmetic statement about weights into a contradiction involving the age of a specific k, and it applies recent lattice-simplex bounds in a nontrivial way. The distribution results and conjectures in Section 4 may also be of interest. However, the manuscript currently has several load-bearing gaps and at least one false statement, so the main theorem is not yet established as written.

major comments (5)
  1. [§1, Theorem 1.2] Theorem 1.2 as stated is false for n=1: the only well-formed 1-dimensional weighted projective space is P^1, which has Fano index 2, while y_1(2y_1−1)=1. The proof in Section 3 explicitly assumes n≥4, so the statement in the abstract and in Theorem 1.2 must be qualified (for example, n≥2) and the low-dimensional cases n=2,3 must be either proved or explicitly relegated to prior results, as is done for n=3 in the introduction.
  2. [§3, Lemma 3.1, around Eq. (3.3)] The displayed inequality "4y_{n−1}^2/(y_n(2y_n−1)) < 1/(4y_n−1)" is false. For n=4, for example, y_3=6 and y_4=42, so the left-hand side is 144/3486 ≈ 0.0413, while the right-hand side is 1/167 ≈ 0.0060. The chain leading to (3.3) therefore does not justify β_n < 1/(4y_n−1). The argument can be repaired: the preceding bound gives β_n < 4y_{n−1}^2/(y_n(2y_n−1)) < 1/(4y_{n−1}), and this is still enough for the subsequent claims 2β_n < 1/y_{n−1} and for the monotonicity of f on (0,1/(4y_{n−1})). The authors should correct (3.3) and the accompanying text.
  3. [§2.4, Lemma 2.11 and §3, Lemma 3.3] The partial-sum bounds x_1+...+x_k ≤ 1/s_1+...+1/s_k in Lemma 2.11 and the analogous bounds in Lemma 3.3 are obtained by citing [2, Lemma 4.5], but that lemma is never stated in the paper. This is a load-bearing step: it provides the majorization that yields the product lower bound in Lemma 2.11 and the majorization of β by the explicit vector z in Lemma 3.3. The authors should either state [2, Lemma 4.5] and verify that its hypotheses are satisfied in the t-shifted setting (noting, for instance, that after substitution the hypothesis reduces to x_1...x_j + S_j ≤ 1), or give a self-contained proof. Without this, the final age contradiction cannot be independently checked.
  4. [§2.3, Lemma 2.10] The proof of Lemma 2.10 is only a sketch: it relies on [2, Theorem 4.3], [2, Lemma 4.2(a)], [2, Claim 4.3.1], and [2, Claim 4.3.3] without stating these results, and the monotonicity check for the function g(l) is delegated to [2, Claim 4.3.3] for l≥4. Since this lemma supplies the lower bound on β_1...β_{n−1} used in Lemma 3.1, the relevant statements and their applicability should be made explicit.
  5. [§4, Theorem 4.4] Theorem 4.4 is a computer-assisted result, but no code, input, or output list is provided. The statement that one can verify the result using the Graded Ring Database [4] or by a computer search that took 155 hours is not a reproducible proof. The authors should provide the search program and the resulting list of Fano indices, or a verifiable certificate, so that the theorem and the conjectures built on it can be checked.
minor comments (5)
  1. [§2.2] The name "Ried" should be "Reid".
  2. [§4 (header)] The section title "Distribution of F ano indices" contains a stray space; it should read "Fano".
  3. [§1, Abstract and Theorem 1.2] The abstract and theorem should state the intended dimension range explicitly, because the n=1 case is a counterexample and the proof in Section 3 starts with "dimension n≥4".
  4. [§3, final paragraph] The phrase "Lemmas 3.1-3.3 provides" should be "provide".
  5. [§2.2, Proposition 2.4] The phrase "for any 2≤k≤q−2" would more naturally be "for every 2≤k≤q−2".

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the Fano-index bound is derived from independent prior bounds on barycentric coordinates and Kasprzyk's age criterion, with self-citations only contextual.

full rationale

The proof of Theorem 1.2 does not assume the target bound. It begins with the elementary identity q(X)=a_1+...+a_{n+1} for well-formed weighted projective spaces, then translates the geometry into an integral simplex with barycentric coordinates beta_i=a_i/q. The subsequent argument uses external, non-self-cited results: the product-sum inequalities of Averkov and Pikhurko (Theorems 2.7 and 2.8), the lower bound beta_n >= 1/(2y_n) from [2], and Kasprzyk's age criterion (Proposition 2.4). The value y_n(2y_n-1) enters only as the threshold in a contradiction argument: assuming q>y_n(2y_n-1), Lemmas 3.1-3.3 derive bounds on a_n, a_{n+1}, and a_i, and the final paragraph uses these bounds to force age(2y_n-1) >= n, contradicting Proposition 2.4. None of these bounds is fitted from data or defined in terms of the claimed conclusion. The reliance on [2, Lemma 4.5] and [2, Theorem 4.3] is a reliance on external prior work, not on the paper's own conclusions; even if those lemmas are unstated and may pose a correctness risk, that is not circularity. The self-citations in the paper are contextual: [14] is cited for the dimension-3 verification of the conjecture and [21] for the low-dimensional distribution coincidence, neither of which supports the n>=4 derivation of Theorem 1.2. The distribution results in Section 4 are obtained by computer enumeration against explicit arithmetic conditions, not by renaming an assumed conclusion. No step reduces by construction to an input, so the appropriate finding is no significant circularity.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters are fitted; the Sylvester sequence is fixed and no constants are tuned to data. The main unsupported inputs are external theorems from [2], especially Lemma 4.5 and Theorem 4.3, which are cited but not reproduced. There are no invented entities.

assumptions (6)
  • standard math Karamata's inequality for the concave function log (Lemma 2.12).
    Stated and proved in the paper; converts majorization into product inequalities in Lemmas 2.11 and 3.3.
  • domain assumption Kasprzyk's age criterion: X has canonical singularities iff age(k) is not equal to n for 2<=k<=q-2 (Proposition 2.4).
    Cited from [18, Proposition 2.5]; it is the contradiction target in the proof of Theorem 1.2 and a filter in the dimension-4 search.
  • domain assumption The integral-simplex correspondence: a well-formed weighted projective space yields an integral simplex with a unique interior point and barycentric coordinates beta_i=a_i/q (Section 2.3).
    Quoted from [3], [6], [18]; it turns weight bounds into bounds on barycentric coordinates.
  • domain assumption Averkov-Kruempelmann-Nill [2, Lemma 4.5]: product-sum inequalities imply partial sums x_1+...+x_k<=1/s_1+...+1/s_k.
    Used, without statement, in Lemma 2.11 and Lemma 3.3 to build the majorizing vector. The lemma's exact hypotheses, especially the tail parameter t>0, are not reproduced.
  • domain assumption Averkov-Kruempelmann-Nill [2, Theorem 4.3] and [2, Lemma 4.2(a)]: structure of Izhboldin-Kurliandchik minimizing solutions for products.
    Used in Lemma 2.10 to derive the lower bound beta_1...beta_{n-1}; the optimization argument is only sketched.
  • domain assumption External simplex bounds: Theorem 2.8 (beta_1...beta_n<=1/q) and Theorem 2.9 (beta_n>=1/(2y_n)).
    Attributed to Pikhurko [27] and Averkov-Kruempelmann-Nill [2]; used throughout Section 3.

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Pith. "Pith review of On Fano indices of weighted projective spaces." pith.science (2026). https://pith.science/paper/FUCBZLGO

@misc{pith2026260803434,
  author       = {Pith},
  title        = {Pith review of: On Fano indices of weighted projective spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FUCBZLGO}},
  note         = {Machine review of arXiv:2608.03434}
}
abstract

The Sylvester sequence is defined recursively by $s_1=2$ and $s_i=s_{1}\cdots s_{i-1}+1$. In this paper, we prove that the Fano index of an $n$-dimensional well-formed weighted projective space with canonical singularities is bounded above by \[ (s_n-1)(2s_n-3). \] This gives an affirmative answer to a conjecture of Chengxi Wang for weighted projective spaces and $\mathbb Q$-factorial toric Fano varieties with Picard number one. We also investigate the distribution of Fano indices among $4$-dimensional weighted projective spaces. As the distribution of Fano indices of weighted projective spaces coincides with that of indices of terminal Calabi--Yau varieties in dimension $n\leq 3$, we expect this coincidence to persist also in dimension 4, and more generally, in all dimensions.

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