{"id":"dda209ed-94ee-41ba-9fa8-8d01d5550a78","arxiv_id":"2510.08089","paper_version":2,"verdict":"REJECT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"A proposed new invariant of the negative part of a big divisor is used to state Noether-type volume bounds, but a false lemma breaks the derivation.","lead":"This paper introduces a number attached to the negative part of a big divisor on an algebraic surface and uses it to prove lower bounds on the divisor's volume from the size of its section space. It aims to unify classical Noether inequalities and extend them to foliations, but the main proof contains invalid algebraic steps.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Reader's counterexample to Lemma 3.3(ii) uses a non-integral fibre class; in the application F is an integral general fibre, so the lemma's equality E_A=E_F is valid and the main theorems stand. No load-bearing flaw identified.","rationale":"The reader's rejection rests on a counterexample to Lemma 3.3(ii) that assumes F is a Q-divisor with fractional intersection F·Γ=1/2. In the proof of Proposition 4.6, F is a general fibre of a fibration and therefore an integral divisor; its intersection with every curve is an integer. For integer intersections, the identities min{1,nF·Γ}=min{1,F·Γ} and e_{nF}=e_F hold, so the disputed step is valid in the application. Furthermore, Proposition 4.6's equality can be proven by a short independent computation from the definitions, so the main Theorems 1.1 and 1.2 are not actually unsupported. I found no other load-bearing error: Lemma 4.2's algebra is correct under the intended integrality assumptions, the volume bounds in Propositions 5.2–5.6 follow from the stated inequalities, and the equality cases are consistent with the derived identities. The paper has genuine presentation weaknesses: Lemma 3.3 should state that F is integral, Lemma 3.5's proof is sloppy as written, and Proposition 4.6 asserts equality without displaying the direct derivation. These merit a conditional acceptance rather than a rejection: the central claims are correct but the write-up needs clarification. A concrete verification of the integrality step or a direct proof of (4.9) would settle the matter definitively.","tokens_in":13497,"tokens_out":40063,"duration_ms":306079,"concrete_test":"Verify the disputed step in the actual configuration of Proposition 4.6: take the fibration f:X→B with general fibre F and compute F·Γ_i for a component Γ_i of the negative part N. Since F and Γ_i are integral curves, F·Γ_i is an integer; check that for every integer x≥0, min{1,nx}=min{1,x} for all n≥1, so E_{nF}=E_F. Alternatively, re-derive Eq. (4.9) directly from the identities M*=M+A, Z=Z*+C, C=A+N without invoking Lemma 3.3(ii); if (4.9) follows, the main theorem is independent of the disputed lemma.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The reader's weakest_assumption targets Lemma 3.3(ii), claiming that E_A=E_F and e_A=e_F fail when n>1 and 0<F·Γ_i<1. This is correct for general Q-divisors, but it does not apply to the only place where the lemma is used, Proposition 4.6. There, F is a general fibre of a fibration, hence an integral divisor. For every integral curve Γ_i in the support of N, F·Γ_i is a nonnegative integer. If F·Γ_i=0, both sides are 0; if F·Γ_i≥1, then min{1, nF·Γ_i}=min{1, F·Γ_i}=1 for every n≥1. Consequently E_{nF}=E_F and e_{nF}=e_F exactly as the proof asserts. The reader's example with F·Γ=1/2 cannot be realized with an integral general fibre on a smooth projective surface. Moreover, even if Lemma 3.3(ii) were stated for Q-divisors, Proposition 4.6's equality (4.9) can be obtained directly from the tautological identities M*=M+A, Z=Z*+C, and C=A+N, so the central inequality does not depend on the disputed interpretation. The manuscript would benefit from stating explicitly that A and F in Lemma 3.3 are integral divisors and from tightening the proof of Lemma 3.5, but these are presentation issues, not a collapse of Theorems 1.1–1.2.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":13911,"tokens_out":28281,"duration_ms":235058,"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":[{"comment":"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.","section":"Lemma 3.3(ii)"},{"comment":"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.","section":"Equation (4.1) and Lemma 4.1(3)"},{"comment":"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.","section":"Definition 3.2 / Lemma 3.5"}],"minor_comments":[{"comment":"The title contains a typo: “SURF ACES” should be “SURFACES”.","section":"Title/Abstract"},{"comment":"“Since K_F ≡ M+Z” appears to be a typo; it should be “Since D ≡ M+Z”.","section":"Section 4, before Lemma 4.1"},{"comment":"The sentence “The inequality (4.9) has proved before” should read “The equality (4.9) has been proved before.”","section":"Proposition 4.6 proof"},{"comment":"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.","section":"Proposition 4.6 equality case"},{"comment":"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.","section":"Notation"}],"recommendation":"major_revision","confidential_remarks":"The reader's principal counterexample to Lemma 3.3(ii) does not invalidate the main theorems, because in the application F is an integral general fibre. However, the manuscript as printed contains a false lemma statement and a sign inconsistency in the definition of Z*; both are local but load-bearing. I recommend major revision rather than reject, since the central argument appears salvageable with explicit integrality hypotheses and corrected signs. I would be willing to review a revised version."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe main results of this paper are probably right, but the reader's instinct was not baseless: Lemma 3.3(ii) as stated is false. The saving grace is that the only application of that lemma, in Proposition 4.6, works because the divisor F is an integral general fibre.\n\nWhat's new: the invariant e(D), a maximum over N-nef divisors of A·N/(A·E_A), is a clean way to package the negative part of the Zariski decomposition. Theorems 1.1 and 1.2 genuinely extend the Noether-type bounds of Tsunoda–Zhang, Shin, and Lü–Tan, and the foliated applications, including the ps-index inequality, are new. The paper is clearly written and the references look appropriate.\n\nThe soft spot: Lemma 3.3(ii) claims that if A ≡_N nF then E_A = E_F and e_A = e_F. That requires min{1, nF·Γ_i} = min{1, F·Γ_i} for every component Γ_i of N. The reader's (-2)-curve example with F·Γ = 1/2 shows why the general Q-divisor statement fails. But the stress-test is also correct: in Proposition 4.6, F is a fibre of a fibration, hence integral, so F·Γ_i is a nonnegative integer and the equality holds. The same integrality issue appears in the proof of Lemma 3.5, which uses A·E_A ≥ Σ A·Γ_i/(−Γ_i^2); that inequality is valid when A·Γ_i is an integer, but not in general. These are not cosmetic: the lemmas in Section 3 need to be restated with explicit integrality hypotheses. However, I do not see a load-bearing flaw in the main theorems. The reader's objection to Lemma 4.2 is off target: (M*)^2 = M·M* is correct because M* − M is N-supported and M* is N-orthogonal.\n\nBottom line: the paper deserves a serious referee. It needs revision, not rejection. The author should tighten the statements and proofs in Section 3 and make the integrality of F explicit in the use of Lemma 3.3. If that lands, the main inequalities and the foliated corollaries stand.\n\nRecommendation: send it to peer review.","headline":"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.","tokens_in":14330,"tokens_out":12554,"would_cite":true,"duration_ms":88213,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14J29","14D06","32S65"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["Noether inequality","big divisors","Zariski decomposition","volume of divisors","pluricanonical systems","foliated surfaces","ps-index","algebraic surfaces"],"falsifier":"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.","tokens_in":13414,"feed_emoji":"📐","tokens_out":6623,"duration_ms":52759,"temperature":0.7,"pith_summary":"The paper proposes a single numerical invariant e(D), computed only from the negative part N of the Zariski decomposition of a big divisor D on a smooth projective surface, and proves that this invariant controls how far the volume Vol(D) can drop relative to the dimension h0(D) of sections. If D is nef, e(D)=0, and the new inequalities recover classical Noether-type bounds. In the general case, the correction terms in the volume bounds are explicit rational functions of e(D), with equality cases described in terms of rational surfaces of low degree. As applications, the paper derives log Noether inequalities and new lower bounds for the volume of canonical foliated surfaces. This matters because it unifies several previously separate inequalities and identifies the negative part as the single quantity that governs them.","feed_headline":"New invariant governs volume bounds on surfaces","feed_subtitle":"A single number attached to a divisor's negative part yields sharp Noether-type inequalities and extends to foliations.","key_machinery":"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.","core_discovery":"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","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["Negative part invariant tightens surface volume bounds","New divisor invariant sharpens Noether inequalities","Zariski negative part gives sharp volume bounds","Invariant yields sharp volume bounds on surfaces"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Negative part invariant tightens surface volume bounds","New divisor invariant sharpens Noether inequalities","Zariski negative part gives sharp volume bounds","Invariant yields sharp volume bounds on surfaces"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000195,"raw_usage":{"total_tokens":1199,"prompt_tokens":757,"completion_tokens":442,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":501,"completion_tokens_details":{"reasoning_tokens":386}},"tokens_in":501,"tokens_out":442,"duration_ms":4468,"temperature":1.0,"reasoning_tokens":386,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T10:50:18.178101+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}