REVIEW 3 major objections 5 minor 11 references
Noether-type inequalities for big divisors via control of the negative part
T0 review · 3 major / 5 minor · reviewed 2026-08-04 · deepseek-v4-flash
Pith's one-line read This paper introduces a numerical invariant e(D), read off from the negative part of a big divisor's Zariski decomposition, and proves that it controls lower bounds for the divisor's volume in terms of its number of sections.
desk verdict The e(D) invariant is a genuine organizing idea for Noether-type inequalities on surfaces; the reader's counterexample hits overgeneralized lemmas, not the main argument, so the paper is worth refereeing rather than rejecting. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The key new object is the numerical invariant e(D), defined as the supremum of ratios A·N / A·E_A, where A ranges over divisors non-negative on each component of N, and E_A is the effective divisor supported on N satisfying E_A·Γ_i = -min{1, A·Γ_i}. A companion identity, Lemma 3.3, asserts that (e_A E(A) - N)·A ≥ 0 and, when A ≡_N nF, that (e_A/n E(A) - N)·A ≥ 0; this inequality is what converts intersection numbers into volume bounds. In the pencil case the same inequality is applied with A equal to the movable part M, yielding the rational bound (n²/(n+e_M))·DF that drives Proposition 4.6.
What would settle it
Work out Lemma 3.3(ii) on a surface containing a (-2)-curve Γ: take N=Γ, F an N-nef divisor with F·Γ=1/2, and A an N-nef divisor with A·Γ=1. Compute whether (e_A/2)E(A)-N has non-negative intersection with A; a negative value contradicts the lemma and would invalidate Proposition 4.6 and hence Theorems 1.1-1.2.
Extended reading notes
Core claim
The central claim is Theorems 1.1 and 1.2: for a big integral divisor D with no D-exceptional curves, h0(D)≥2, the Zariski decomposition D=P+N, the invariant e(D)=max_A (A·N)/(A·E_A) over N-nef divisors A (with E_A defined by E_A·Γ_i = -min{1,A·Γ_i}) satisfies Vol(D) ≥ (h0(D)-1)^2/(h0(D)-1+e) when |D| is composed with a pencil, and Vol(D) ≥ h0(D)-2, or Vol(D) ≥ 2h0(D)-4 when X is not ruled, in the non-pencil case. The inequalities are sharp, and equality forces the map induced by |D| to land on a rational surface of minimal degree, or on a double cover of one; additionally, refined corrections apply when equality fails. The same invariant specializes to e=0 for nef divisors, recovering class
Load-bearing premise
The volume bounds rest on Lemma 3.3(ii), which asserts that when a divisor A is numerically equivalent to a positive multiple of F along the negative part N, the associated invariant and correction divisor for A are just scaled versions of those for F; if that assertion fails, the main inequalities do not follow.
Editorial extensions
If this is right
- For nef and big D, the invariant vanishes, so Theorem 1.2 reduces to Vol(D) ≥ h0(D)-2, and to Vol(D) ≥ 2h0(D)-4 when X is not ruled, recovering classical Noether-type inequalities for surfaces.
- For D = m(K_X+Δ), the invariant is bounded by 2m, so the logarithmic plurigenera satisfy explicit inequalities of Noether type, extending the classical bounds to non-complete surfaces of general type.
- For a canonical foliated surface F, the invariant satisfies e(mK_F) ≤ m, yielding lower bounds for Vol(F) in terms of the plurigenera P_m(F) in both the pencil and non-pencil cases.
- A direct corollary is the ps-index bound Vol(F) ≥ 1/(λ(F)²(1+λ(F))), a new effective lower bound for foliated surfaces of general type.
- Equality cases in the main theorems are classified as morphisms onto surfaces of degree d-1 or 2d-2 in projective space, connecting the inequalities to the classical classification of minimal-degree surfaces.
Reading between the lines
- If the main inequalities hold, the invariant e(D) could serve as a quantitative measure of how far a divisor is from being nef, since it is the sole correction term in an otherwise uniform volume bound.
- The framework suggests a testable extension to higher-dimensional log pairs by replacing the negative part with the divisorial part of a Zariski decomposition; the surface-specific Hodge-index arguments would need a substitute.
- The equality classifications imply that surfaces attaining the sharp bounds form a very short list; checking that list against explicit examples of foliated surfaces would test the sharpness of the foliation bounds.
- Because the proof of Lemma 3.3(ii) is not established in the paper, the volume bounds as stated may require a repaired argument; if the lemma fails, weaker versions of the inequalities may still hold under extra hypotheses on the intersections A·Γ_i.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper proposes a new numerical invariant e(D) attached to the negative part N of the Zariski decomposition of a big integral divisor D on a smooth projective surface, and uses it to prove Noether-type lower bounds for vol(D) in terms of h^0(D). The two main theorems give a pencil-case bound vol(D) >= (h^0(D)-1)^2 / (h^0(D)-1+e) and non-pencil bounds vol(D) >= h^0(D)-2 and, for non-ruled X, vol(D) >= 2h^0(D)-4, together with refined inequalities and equality characterizations. The paper also applies these results to log surfaces and canonical foliations, claiming to recover results of Tsunoda-Zhang, Shin, and Lü-Tan.
Significance. If the results are correct, the invariant e(D) provides a unified and explicit mechanism for volume-H^0 inequalities on surfaces, with applications to adjoint divisors and foliated surfaces. The main inequalities are sharp in classical cases and the correction terms are expressed solely through the negative part N. The paper does not rely on fitting parameters to the target inequalities, and the applications to foliations are a useful addition. However, the manuscript as printed contains a false lemma statement and an inconsistent definition of Z*, so the proof needs careful correction before the claims can be accepted.
major comments (3)
- [Lemma 3.3(ii)] Lemma 3.3(ii) is false as stated if F is allowed to be a Q-divisor. The proof asserts E_A = E_F and e_A = e_F from A ≡_N nF. For a (-2)-curve Γ, N=Γ, F·Γ=1/2, A=2F, n=2, one has E_A = (1/2)Γ, E_F=(1/4)Γ, e_A=2, e_F=4, and ((e_A/2)E(A)-N)·A = -1/2 < 0. In the only application, after (4.7), F is an integral general fibre of a fibration, so F·Γ_i is a nonnegative integer and the step E_{nF}=E_F is valid. The lemma and Definition 3.2 must explicitly state that A and F are integral divisors; otherwise the printed statement is false and the proof of the lemma is invalid.
- [Equation (4.1) and Lemma 4.1(3)] As printed, (4.1) defines Z^* := Z + Σ_i c_i Γ_i with Z^*·Γ_i = 0. With Γ_i^2<0, this forces c_i = -(Z·Γ_i)/Γ_i^2 ≥ 0, so Z^* ≥ Z. This contradicts Lemma 4.1(3), which asserts Z - Z^* = N + (M^*-M) ≥ 0, and it makes the identity P ≡ M^*+Z^* used in Lemma 4.2 and Proposition 4.6 false. The intended definition must be Z^* := Z - Σ_i c_iΓ_i (with c_i≥0), or equivalently the coefficients in (4.1) must be chosen with the opposite sign. Since the whole volume estimate depends on P = M^*+Z^*, this sign issue is load-bearing and must be corrected.
- [Definition 3.2 / Lemma 3.5] The definition of e(D) as a maximum over N-nef divisors is not well-posed if Q-divisors are allowed. Lemma 3.5 proves finiteness only under the integrality assumption that A·Γ_i is either 0 or at least 1 for each component Γ_i of N. Without integrality, one can take A = εF with F·Γ=1/2 on a (-2)-curve Γ; then e_A ~ 4/ε → ∞, so the supremum is infinite. The text should state in Definitions 3.2 and 3.4 that A ranges over integral N-nef divisors, and Lemma 3.5 should repeat this hypothesis. This is not just a technicality: it is needed for the invariant e(D) to be finite and for the proof of Lemma 3.5 to be valid.
minor comments (5)
- [Title/Abstract] The title contains a typo: “SURF ACES” should be “SURFACES”.
- [Section 4, before Lemma 4.1] “Since K_F ≡ M+Z” appears to be a typo; it should be “Since D ≡ M+Z”.
- [Proposition 4.6 proof] The sentence “The inequality (4.9) has proved before” should read “The equality (4.9) has been proved before.”
- [Proposition 4.6 equality case] The proof asserts without justification that under DF=1 and Supp(Z)=Supp(N) one has (e_M(F^*-F)-N)·F=0. This is an equality case of Lemma 3.3 and should be proved explicitly, since it underlies the equality characterization in Theorem 1.1.
- [Notation] The use of e(D), e_M, e_A, and e_0 is sometimes confusing. In particular, Theorem 1.1 writes e in the denominator while Proposition 5.3 uses e_M; the inequality e_M ≤ e is stated in the proof but should be stated in the theorem or proposition for clarity.
Circularity Check
No significant circularity: the invariant e(D) is constructed from the negative part N, and the main inequalities are derived independently of any fitted quantity or self-citation chain.
full rationale
I find no load-bearing circular step. The central invariant e(D) is defined directly from the negative part N of the Zariski decomposition as a maximum over N-nef divisors A of A.N/(A.E_A), and Lemma 3.5 bounds it by e0 = max gamma_i(-Gamma_i^2) without reference to vol(D) or h^0(D). The main inequalities are then proved from the Zariski decomposition, the movable/fixed part decomposition, Hodge index theorem, and standard degree bounds for surfaces in projective space. No parameter is fitted to the target inequalities and then renamed as a prediction. The external citations used in the appendices, such as [TZ92], [Shin08], [Bru15], and [McQ08], are used either to bound the input e in special cases or to compare/complement the results; they do not supply the target inequalities themselves. The citation to [LT24] is for comparison and sharpness examples, not for the proof of the main theorems. The mathematical concern raised about Lemma 3.3(ii) is a potential correctness issue, not a circularity: even if the interchange E_A=E_F were invalid in some generality, that would not make the theorem equal to its assumptions by construction. In the application of that lemma, F is an integral general fibre of a fibration, so the disputed fractional-intersection example does not occur. The acknowledgement of referee corrections mentions errors but does not identify any circular reduction. Overall, the derivation chain is self-contained with respect to the claimed predictions, and the score is 0.
Assumptions & free parameters
assumptions (6)
- standard math Existence and uniqueness of the Zariski decomposition for pseudo-effective divisors on smooth projective surfaces.
- standard math Negative-definite intersection matrices imply positivity of effective divisors with nonpositive intersections (Lemma 2.4).
- standard math Hodge index theorem for surfaces.
- standard math Beauville-Nagata degree bounds and classification of low-degree surfaces in projective space.
- domain assumption For canonical foliations, the negative part of K_F is a disjoint union of maximal F-chains (Brunella/McQuillan).
- domain assumption One may contract all D-exceptional curves without changing h0(D) or Vol(D).
Cite this review
Pith. "Pith review of Noether-type inequalities for big divisors via control of the negative part." pith.science (2026). https://pith.science/paper/HWJEGTV4
@misc{pith2026251008089,
author = {Pith},
title = {Pith review of: Noether-type inequalities for big divisors via control of the negative part},
year = {2026},
howpublished = {\url{https://pith.science/paper/HWJEGTV4}},
note = {Machine review of arXiv:2510.08089}
}
abstract
Let $X$ be a smooth projective surface over $\mathbb{C}$ and $D$ a big divisor with Zariski decomposition $D=P+N$. We study the relationship between the volume $\mathrm{vol}(D)=P^2$ and the dimension $h^0(D)$. We introduce a numerical invariant $\mathfrak{C}(N)$ depending only on the negative part $N$, which provides a universal baseline control for $\mathrm{vol}(D)$. This allows us to establish Noether-type inequalities relating $\mathrm{vol}(D)$ and $h^0(D)$, where all correction terms are explicitly governed by $\mathfrak{C}(N)$. Our results recover and unify several classical inequalities on surfaces, and apply in particular to adjoint divisors and foliations. We further obtain lower bounds for $\mathrm{vol}(D)$ in terms of the ps-index $\iota(D)$, with applications to foliated surfaces.
Reference graph
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