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Counting points on a weighted projective space, the leading exponent is the sum of the weights, and the constant carries an explicit gcd(q, φ(me))^{-1} arithmetic sparsity factor.

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The claimed asymptotic for weighted projective spaces, with a gcd(q, φ(me))^(-1) sparsity factor, is not established because the reduction to projective point counts ignores the Veronese map's non-surjectivity.

T0 review reviewed 2026-08-05 challenge →

load-bearing objection The rational-point count is fine; the fixed-degree theorem is unsupported: Prop 4 misuses Widmer's exponent and the sparsity factor is introduced by definition. the 4 major comments →

arxiv 2509.02319 v3 pith:B4XD5KOY submitted 2025-09-02 math.NT math.AG

Arithmetic Sparsity and Obstructions in Weighted Projective Spaces

Tanush Shaska This is my paper
classification math.NT math.AG MSC 11G5011G3514G0514M25
keywords weighted projective spacesweighted heightsarithmetic sparsityKummer torsorsVeronese morphismfixed-degree algebraic pointsBatyrev–Manin conjecturepoint counting asymptotics
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

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 counts rational and algebraic points of bounded height on weighted projective spaces, where weights q=(q0,...,qn) scale the coordinates. Its central claim is that the count follows a power law: over a degree-m number field, roughly D X^{meQ} points of degree e, with Q=q0+...+qn, and over Q with e=1, 2^n/ζ(Q) X^Q. The constant D is the novel part: compared with classical projective-space counts, it is divided by the Veronese degree and carries an extra factor gcd(q, φ(me))^{-1}, which the paper reads as the density of points whose preimage under the Veronese map exists without a field extension. The paper interprets this factor cohomologically as the triviality of Kummer torsors, analogous to a Brauer-group obstruction. The abstract also claims a stronger form over Q in which the exponent itself is the value of a linear program over a valuation set M_w, so the exponent can differ from and even exceed Q.

Core claim

The central claim is that arithmetic sparsity in weighted projective spaces is encoded in the leading constant of a counting asymptotic, via the Veronese morphism that raises each coordinate to the power q/q_i. Over Q the body proves Z_h(WP^n_q(Q),X)=2^n/ζ(Q) X^Q+O(X^{Q-min q_i}); for degree-e points over a degree-m field with n>e it proves Z_h(WP^n_q(k;e),X)=D X^{meQ}+O(X^{meQ - min q_i/m} log X), where Q=Σq_i and D is a sum over extensions of terms proportional to |Δ_K|^{(n+1)/2} gcd(q,φ(me))^{-1}, divided by the Veronese degree. The gcd factor is the paper's arithmetic obstruction: the density of points in P^n(k) whose Kummer torsor is trivial. The abstract states a stronger result over Q

What carries the argument

The load-bearing mechanism is the Veronese morphism together with the normalized weighted height h(p)=H(φ(p))^{1/q}, which converts weighted-height counts into projective-space counts with a finite preimage structure of degree q^n/∏q_i. In the body, counting uses weighted-gcd normalization, Möbius inversion, and the cited fixed-degree projective asymptotics, producing the factor gcd(q,φ(me))^{-1}; cohomologically, this factor is the density of trivial Kummer torsors under ∏_i μ_{q/q_i}. In the abstract's formulation, the same obstruction is encoded in a valuation set M_w of Kummer congruences, and the exponent is the value of a linear program over its minimal elements.

Load-bearing premise

The counting argument assumes the weighted-height count follows the same leading power law as the projective count after dividing by the Veronese degree and the obstruction factor; if the Veronese map's failure to be surjective changes the exponent rather than only the constant, the asymptotic in Theorem 3 does not follow.

What would settle it

Enumerate wgcd-normalized integer triples for WP^2_{(1,2,3)}(Q) with |x0|≤X, |x1|≤X^2, |x2|≤X^3. Theorem 1 predicts the count is 4/ζ(6) X^6 + O(X^3), so a log-log fit for X=10^2...10^5 should have slope 6 and intercept about 3.93; if the fitted slope differs, or if the abstract's linear-program exponent 6a(w) is measurably different from 6, the two statements in the paper can be separated.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • Over Q, rational points of bounded weighted height on WP^n_q follow 2^n/ζ(Q) X^Q, so Q=q0+...+qn is the exact leading exponent for the rational count.
  • For points of degree e over a degree-m field, the count is D X^{meQ}, and the leading constant is smaller than the projective-space constant by gcd(q, φ(me))^{-1} and by the Veronese degree: the sparsity is in the main term, not the error.
  • A point of P^n(k) lifts to WP^n_q(k) only when its Kummer torsor is trivial; the paper shows this condition fails for explicit rational points, so the Veronese map is non-surjective on rational points.
  • The relation between projective and weighted counts in the paper implies that the obstruction factor is invisible for points defined over Q itself (gcd(q, φ(1))=1) and appears only for algebraic points of degree e>1.
  • If the abstract's linear-program result holds, the exponent can be larger than Q, so weighted projective spaces can produce growth rates that are not just rescaled projective rates.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the abstract's linear-program result is correct, then the body's Theorems 1–3 describe only the cases where the LP optimum equals Q; for weights such as (1,2,3) the predicted exponents conflict, and a finite enumeration of normalized triples can decide which statement governs.
  • The cohomological density interpretation suggests the same gcd(q, φ(me))^{-1} factor should appear for any finite morphism whose fibers are torsors under roots of unity, so the counting method likely transfers to weighted complete intersections and quotient stacks.
  • The weight-dependent exponent claimed in the abstract, if true, would make weighted projective spaces a test bed for the weighted analogue of the Batyrev–Manin conjecture: the linear-program value a(w) would play the role usually played by the Fujita invariant.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper studies rational and algebraic points of bounded weighted height on weighted projective spaces WP^n_q(k). It proves by lattice-point counting a Schanuel-type asymptotic over Q (Theorem 1) and over a general number field k (Theorem 2), with leading exponent mQ, Q = q_0+...+q_n. It then claims, for fixed-degree algebraic points, an asymptotic Z_h(WP^n_q(k;e),X) ~ D X^{meQ} with D containing a factor gcd(q,φ(me))^{-1} (Theorem 3), interpreting this factor as an arithmetic obstruction to the surjectivity of the Veronese morphism φ. The abstract and introduction additionally advertise a main result over Q with exponent q a(w) and a linear program over a valuation set M_w, but this result does not appear in the body.

Significance. If the fixed-degree asymptotic were correct, it would be a substantial extension of Schanuel--Schmidt--Widmer counting and would introduce an interesting arithmetic-sparsity phenomenon. The rational-point lattice counts in Theorems 1 are a natural and plausible extension of Schanuel's method to the weighted height. However, the central claim of the paper, Theorem 3, is unsupported: its proof rests on an invalid application of Widmer's theorem and on a circular introduction of the gcd factor. Because the resulting exponent contradicts the paper's own rational-point theorem, the advertised main result is not established.

major comments (4)
  1. [Proposition 4, Eqs. (17)–(18)] The reduction to Widmer fails twice. First, Eq. (17) identifies d_φ^{-1} Z_H(φ(WP^n_q(K/k)), X^q) with Z_h(WP^n_q(K/k), X), but φ(WP^n_q(K/k)) is generally a proper subset of P^n(K/k). For q=(1,2,3), n=2, the point [1:2:3] ∈ P^2(Q) has no rational preimage: a preimage would require a rational λ with v_2(λ) divisible by 6 (so λ is a sixth power) and v_2(λ)+1 divisible by 3 (so 2λ is a cube), which is impossible. Thus the target-space count overcounts. Second, Eq. (18) applies Widmer with exponent meQ; Widmer's leading exponent for P^n(K/k) is me(n+1), not meQ. With q=(1,2,3), k=Q, e=1, Eq. (18) gives X^{q·meQ}=X^{36}, while Theorem 1, a direct lattice count of the same Z_h, gives X^{Q}=X^6. Since Proposition 4 is the sole support of Theorem 3, the fixed-degree asymptotic in Theorem 3 is invalid.
  2. [Lemma 5, Theorem 3, Lemma 7] The sparsity factor gcd(q,φ(me))^{-1} is not derived from a count of the image of φ but is written into the definition of V_K in Lemma 5. Lemma 5 simply asserts that wgcd-normalization 'scales the volume by gcd(q,φ(me))^{-1}' with no independent proof. Theorem 3 then reports the same factor as an arithmetic obstruction, and Lemma 7 calls it the density of splittable torsors. This is circular: the factor is an assumption, not a consequence. The relation in Eq. (22) and Proposition 5 likewise assume the leading-constant relation they are supposed to establish, and Eq. (22) is not an identity of counting functions for finite X.
  3. [Abstract and Introduction vs. Sections 3–4] The abstract promises a main theorem over Q with asymptotic X^{q a(w)} P_w(log X), where a(w) and β(w) are the value and dimension of the optimal face of a linear program over a valuation set M_w, and says that proper strata can dominate. None of this appears in the body: M_w is never defined, a(w) and β(w) never occur in Sections 3–4, and the theorems proved there give exponent Q or meQ with no P_w(log X) factor. This is a major gap between the advertised result and the proven statements.
  4. [Lemma 6] Lemma 6 is used in Theorem 3 to ensure convergence of the sum over fields K∈C_e. Its stated hypothesis is n>e, but the computation in the proof does not establish this. Substituting F(1) = e^2+e+2 - e(e-1)(n+1) into the derived convergence condition mF(1)+2<−1 gives, after the paper's own displayed algebra, the bound n>(e^2+e+2+3/(me))/(e-1)−1. For e=2,m=1 this says n>8.5, not n>e. Thus the convergence of D and of the error term under the stated hypothesis n>e is not proved.
minor comments (4)
  1. [Notation throughout] The symbol q denotes both the weight vector (q_0,...,q_n) and its lcm, while Q denotes both the sum q_0+...+q_n in Sections 2–4 and the sum of reciprocals in the Introduction after Eq. (2). This makes the exponent claims very hard to parse.
  2. [Example 4] Example 4 is incorrect: for WP^1_(2,3)(Q) with φ([x_0:x_1])=[x_0^3:x_1^2], the point [1:2] has rational preimage [1/2:1/2]. A correct non-surjectivity example would be [1:2:3] in WP^2_(1,2,3)(Q), as in the major comments.
  3. [Theorem 2] The error term in Theorem 2 is X^{mQ−1/m} log X, which for k=Q gives X^{Q−1} log X, whereas Theorem 1 has error X^{Q−min q_i}. These disagree when min q_i>1; the discrepancy needs a remark or correction.
  4. [Eq. (22)] Eq. (22), Z_H(P^n(k;e),X)=gcd(q,φ(me)) Z_h(WP^n_q(k;e),X^{1/q}), is stated as a relation of counting functions. At face value it is false for finite X; at best it is a leading-constant comparison. It should be labeled as such and not used in Proposition 5 as an exact identity.

Circularity Check

5 steps flagged

The leading exponent and the arithmetic-obstruction factor are inserted at Lemma 5 and Proposition 4, then returned as discovered in Theorem 3 and Lemma 7.

specific steps
  1. self definitional [Section 4.1, Lemma 5]
    "Let VK = 2−sK (n+1)|∆K|(n+1)/2 gcd(q, φ(me))−1. This represents the adjusted covolume of the primitive lattice, incorporating a sparsity factor from the Veronese map. ... Normalization by wgcd(x0, . . . , xn) = 1 restricts to primitive points, scaling the volume by gcd(q, φ(me))−1. This factor reflects the weighted action under the Veronese map ϕ, where preimages involve roots of unity of order dividing q in fields of degree me over Q, reducing the effective number of rational lifts [2]."

    The quantity that Theorem 3 and Lemma 7 present as the newly found arithmetic obstruction, gcd(q, φ(me))^{-1}, is written into the covolume V_K by definition before any counting. The proof gives no independent computation of the image or of root-of-unity torsors; it merely asserts that wgcd-normalization scales the volume by that factor and cites [2]. Theorem 3 then returns the same factor as part of D, and Lemma 7 renames it as a density. The output is therefore an input by construction.

  2. self definitional [Section 4.2, Proposition 4, Eqs. (17)-(18)]
    "Since h_K(p) = H(ϕ(p))^{1/q}, h_K(p) ≤ X implies H(ϕ(p)) ≤ X^q. Thus (17) Z_h(WP^n_q(K/k), X) = 1/(q^n/∏q_i) Z_H(ϕ(WP^n_q(K/k)), X^q). By Widmer’s [22, Theorem 4.1], adjusted for ϕ with V_K (18) Z_H(P^n(K/k), X^q) = D_K(X^q)^{meQ} + O(B_K(X^q)^{meQ−1}),"

    Equation (17) counts only the image ϕ(WP^n_q(K/k)), a generally proper thin subset of P^n(K/k), but equation (18) silently replaces it by all of P^n(K/k) and assigns the weighted-space exponent meQ to Widmer's projective count. Widmer's exponent for P^n(K/k) would be me(n+1), not meQ. The desired weighted asymptotic is thus imported into the projective theorem, then divided by d_phi; no independent count of the omitted points is made. Combined with V_K containing the gcd factor, this makes the theorem's leading term a restatement of the assumptions.

  3. renaming known result [Section 5.3, Lemma 7]
    "Lemma 7. The constant D from Thm. 3 is given by D∝δ·d_ϕ^{-1}, δ = gcd(q, φ(me))^{-1}, where d_ϕ = q^n/∏q_i is the geometric degree of ϕ, and δ is the density of y∈P^n(k) such that the associated torsor T_y under ∏_i μ_{q/q_i} is trivial. ... The factor δ arises from the proportion of torsors that are trivial over k."

    No computation of this torsor density is supplied. The symbol δ is exactly the factor already inserted in V_K in Lemma 5 and propagated through D_K and D. Calling it a density of trivial torsors is a renaming of the predefined constant, not a derivation. The cohomological language does not provide independent evidence for the factor.

  4. self definitional [Section 6, Eqs. (22)-(24)]
    "The relation (22) Z_H(P^n(k;e), X) = gcd(q, φ(me)) Zh(WP^n_q(k;e), X^{1/q}), adjusts Widmer’s (23) Z_H(P^n(k;e), X) = D^N X^{meQ} + O(X^{meQ−1} log X), to (24) D^N/gcd(q,φ(me)) X^{meQ} + O(...), with our D∝gcd(q,φ(me))^{-1} quantifying the sparsity."

    This relation is asserted without proof, and it is exactly the sparsity statement that Proposition 5 is supposed to establish. If (22) holds, the reduction by gcd(q, φ(me)) is already built into the relation between the two counting functions; Proposition 5's proof, which says 'By (22) and Thm. 3', then concludes a factor already assumed. It is a definition of the claimed sparsity, not an independent verification.

  5. self citation load bearing [Section 4.1, Lemma 5, proof]
    "This factor reflects the weighted action under the Veronese map ϕ, where preimages involve roots of unity of order dividing q in fields of degree me over Q, reducing the effective number of rational lifts [2]."

    The only cited support for the central obstruction factor is [2] (Beshaj–Gutierrez–Shaska), which shares an author with the present paper. That prior work is not machine-checked or otherwise independently verified here, and the claim that the factor 'reduces the effective number of rational lifts' is precisely the content to be proved. The citation is load-bearing because without it the factor in V_K has no derivation.

full rationale

The main asymptotic of the paper (Theorem 3) is not derived from an independent count of the image of the Veronese map. Its two distinctive ingredients—the exponent meQ and the sparsity factor gcd(q, φ(me))^{-1}—are introduced by construction in Proposition 4 and Lemma 5, respectively. Equation (18) assigns the weighted exponent to Widmer's projective count and replaces the proper image ϕ(WP^n_q(K/k)) with the full P^n(K/k), dropping the restriction in Equation (17). Lemma 5 places the gcd factor inside V_K before any counting, and the same factor is later reported as a discovered density of splittable torsors in Lemma 7 and as the content of the asserted relation (22). Because the claimed arithmetic obstruction is an assumption embedded in the covolume, and the claimed leading term is taken from Widmer with the exponent adjusted to the weighted value, the central result reduces to its inputs. This is more than incidental self-citation: the derivation is circular at the point where the asymptotic is formulated.

Axiom & Free-Parameter Ledger

1 free parameters · 3 axioms · 1 invented entities

The central constant of Theorem 3 rests on two unpaid inputs: Eq. (17), which conflates the image of φ with all of P^n, and Lemma 5, which defines the covolume to contain the sparsity factor. The abstract also introduces an M_w framework with no presence in the body. Standard inputs are Schanuel and Widmer theorems.

free parameters (1)
  • Arithmetic sparsity factor gcd(q, φ(me))^(-1) = q = lcm(q_0,...,q_n), m = [k:Q], e = degree
    Introduced by definition in the adjusted covolume V_K (Lemma 5); no counting argument produces it. The factor appears in the theorem only because it was put into the definition.
axioms (3)
  • ad hoc to paper Eq. (17): Z_h(WP^n_q(K/k), X) = (1/d_φ) Z_H((φ-image), X^q), and the image is treated as all of P^n(K/k).
    Needed to reduce weighted counting to projective counting. It assumes φ has full image and uniform fiber multiplicity; the paper's own Example 4 contradicts this, and exponent comparison for weights (1,2,3) shows a power-of-X error.
  • ad hoc to paper Lemma 5: wgcd-normalization scales the primitive-lattice covolume by gcd(q, φ(me))^(-1).
    The lemma asserts the factor without computation; it is the only source of the advertised constant in Theorem 3.
  • standard math Schanuel's theorem for P^n(k) and Widmer's fixed-degree count for P^n(k;e).
    Used as black boxes in Prop. 4 and Theorem 3; they are standard, but the bridge to them (Eq. 17-18) is invalid, and the exponent used is meQ rather than me(n+1).
invented entities (1)
  • Valuation constraint set M_w no independent evidence
    purpose: Defines which valuation vectors in P^n(Q) lift along φ; the abstract's main asymptotic is stated in terms of its minimal elements and a linear program.
    M_w appears only in the metadata abstract. The body never defines it, proves its properties, or gives a theorem using it, so no independent check is possible.

reviewed 2026-08-05 · how reviews work

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Cite this review

Pith. "Pith review of Arithmetic Sparsity and Obstructions in Weighted Projective Spaces." pith.science (2026). https://pith.science/paper/B4XD5KOY

@misc{pith2026250902319,
  author       = {Pith},
  title        = {Pith review of: Arithmetic Sparsity and Obstructions in Weighted Projective Spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/B4XD5KOY}},
  note         = {Machine review of arXiv:2509.02319}
}
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abstract

We study rational and algebraic points of bounded height on weighted projective spaces. A weighted projective space $\mathbb{P}^n_{\mathbf{w}}$, with weights $\mathbf{w} = (q_0, \dots, q_n)$, carries two natural heights: the tautological height $\widetilde{h}$, attached to the tautological bundle on the associated stack, and the weighted height $h = H(\phi(\,\cdot\,))^{1/q}$, the normalized pullback of the Weil height under the Veronese morphism $\phi : \mathbb{P}^n_{\mathbf{w}} \to \mathbb{P}^n$, where $q = \operatorname{lcm}(q_0, \dots, q_n)$. For $\widetilde{h}$ we prove a Schanuel-type theorem over any number field $k$ of degree $m$, with leading term $c_k(\mathbf{w}) X^{mQ}$, where $Q = q_0 + \cdots + q_n$. Our main result concerns $h$ over $\mathbb{Q}$, for arbitrary coprime weights. A point of $\mathbb{P}^n(\mathbb{Q})$ lifts along $\phi$ only if its valuation vector at every prime lies in a set $M_{\mathbf{w}}$ cut out by Kummer congruences. We prove that the points with all coordinates nonzero satisfy \[Z^{\circ}_{h}(\mathbb{P}^n_{\mathbf{w}}(\mathbb{Q}), X) = X^{q\,a(\mathbf{w})} P_{\mathbf{w}}(\log X) + O(X^{q\,a(\mathbf{w}) - \theta})\] for some $\theta > 0$, where $P_{\mathbf{w}}$ has exact degree $\beta(\mathbf{w})$ and positive leading coefficient, and $a(\mathbf{w})$ and $\beta(\mathbf{w})$ are the value and the dimension of the optimal face of a linear program over the minimal elements of $M_{\mathbf{w}}$. The exponent need not equal the projective benchmark $q(n+1)$, and can exceed $Q$. Since the coordinate strata are again weighted projective spaces, the full count follows by stratification, and a proper stratum may dominate. We also count points of fixed degree when the exponents $q/q_i$ are pairwise coprime, and conjecture the asymptotic for $h$ over an arbitrary number field.

Figures

Figures reproduced from arXiv: 2509.02319 by Tanush Shaska.

Figure 1
Figure 1. Figure 1: The weighted ϕ and its extension to the algebraic closure [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

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Cited by 1 Pith paper

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

  1. Graded Keller maps and the Jacobian Conjecture

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    A proposed 3-variable polynomial map with constant Jacobian -2 and generic degree 3 is shown to be a non-invertible Keller map, and graded Keller maps are classified by the sign pattern of their weight vectors.

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This paper was first reviewed by deepseek-v4-flash on August 5, 2026.