{"id":"7c57381e-5656-405d-a52b-c1ee1281f36e","arxiv_id":"2506.05248","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Multiplicities and degree functions of graded families of ideals are expressed as limits of intersection products, with a closed formula for Q-divisorial filtrations in dimension 2.","lead":"This mathematics paper derives formulas that express the growth rate (multiplicity) of a chain of nested ideals using intersection numbers on birational models of a local ring. It also builds examples showing that in two-dimensional rings these formulas can fail for non-Noetherian families.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 4.7's remaining-case estimate uses a reversed sheaf inclusion; as written the lower-bound proof for Theorem D does not close.","rationale":"The reader's conditional verdict is reasonable: the paper depends on an unpublished companion preprint [20] and on structural results from [15], and the reader flagged those. My stress-test looks inside the paper's own argument and finds a more specific, fixable-looking gap in Proposition 4.7, which is the key lower-bound step for Theorem D. The inequality h^0(O_X(-⌈nΔ⌉)⊗E_n^{-1}) ≤ h^0(O_X(-⌈nΔ⌉-rΔ)⊗E_n^{-1}) is reversed under the standard convention that rΔ is effective and O_X(-rΔ) is an ideal sheaf. Because the contradiction in the remaining case depends on this inequality, the proof of Theorem 4.2, and hence Corollary 4.3, is incomplete as written. This does not establish that the theorem is false; it may be repairable by showing the remaining case is vacuous for large n, and the geometric statement is plausible. Therefore I do not move the verdict to reject, but the paper should not be accepted unconditionally until Proposition 4.7 is corrected or replaced. This is why I keep the reader's conditional verdict while noting that the specific weak point I found is internal rather than purely a matter of external dependencies.","tokens_in":25780,"tokens_out":26636,"duration_ms":306624,"concrete_test":"Re-derive the remaining-case estimate with correct sheaf inclusions. Concretely, take C of genus 1, L_C = O_C(np) with n large, and an effective divisor F of degree about n/2; check the claimed Inequality h^0(A) ≤ h^0(A ⊗ O_C(-rδ)) fails when the right-hand line bundle has nonpositive degree while the left-hand one has many sections. Then revisit the chain after equation (19) with the correct inclusion B ⊂ A, and either supply a replacement argument proving deg(E_n) ≤ f, or prove that the case deg(A) ≤ 2p_a(C) - 2 cannot occur for n ≫ 0 using (19) and the linear growth of h^0(A).","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem D rests on Proposition 4.7. In the remaining case of that proof, set A = O_X(-⌈nΔ⌉) ⊗ E_n^{-1} on C and B = A ⊗ O_X(-rΔ) = O_X(-⌈nΔ⌉-rΔ) ⊗ E_n^{-1}. Since rΔ is an effective divisor, O_X(-rΔ) is an ideal sheaf, so B is a subsheaf of A and h^0(B) ≤ h^0(A). The text asserts the opposite inequality, h^0(A) ≤ h^0(B) = χ(B). The subsequent chain e > h^0(L_C) - h^0(A) ≥ χ(L_C) - χ(B) = -deg(O_C(-rΔ)) + deg(E_n) > e depends on that assertion. With the correct inclusion the inequality points the wrong way and the stated contradiction does not follow. The remaining case may in fact be impossible for n ≫ 0, because (19) forces h^0(A) to grow linearly, and a line bundle on a curve with h^0 growing linearly must have degree growing linearly, contradicting deg(A) ≤ 2p_a(C) - 2; however, that argument is not supplied. Thus Proposition 4.7, and therefore Theorem 4.2 and Corollary 4.3, are not proved as written. This is an internal issue independent of the paper's reliance on the companion preprint [20] and on [15].","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies degree functions and multiplicities of graded families of m_R-primary ideals in excellent normal local rings. Using the intersection product developed by the authors in the companion preprint [20], it proves that the multiplicity of a graded family modulo a principal radical ideal can be expressed as a limit of sums of intersection numbers (Theorem B), and that this limit and the sum commute when the union of Rees valuations is finite and the individual limits exist (Corollary C). In dimension 2, for Q-divisorial filtrations, it states a closed formula for the limit of the intersection numbers in terms of a Q-divisor Delta (Theorem D), from which a degree-function formula follows (Corollary E). The paper also constructs two divisorial filtrations on 2-dimensional normal local rings, one with only finitely many Rees valuations overall and one with infinitely many, and an example showing that the limit and sum in Theorem B need not commute in general.","tokens_in":26050,"tokens_out":12015,"duration_ms":134516,"significance":"If the proofs are completed, the paper gives a natural extension of Rees's classical degree-function theory to graded families, with a clean geometric formula in dimension 2. The examples in Section 5 clarify an interesting distinction between Noetherian and non-Noetherian filtrations. The paper is well organized and careful with notation, and it explicitly identifies its dependence on the companion preprint [20] and on [15]. However, the proof of the lower bound in Proposition 4.7 contains a sign error in a key sheaf inclusion, so the central dimension-2 theorem is not proved as written. The remaining parts of the paper are largely coherent, and the gap appears local and likely repairable.","major_comments":[{"comment":"In the 'remaining case' after Eq. (19), the proof asserts h^0(O_X(-⌈nΔ⌉)⊗E_n^{-1}) ≤ h^0(O_X(-⌈nΔ⌉-rΔ)⊗E_n^{-1}) and then uses this inequality in the chain e > h^0(L_C)-h^0(A) ≥ χ(L_C)-χ(B) = -deg(O_C(-rΔ)) + deg(E_n) > e. Since rΔ is an effective divisor, O_X(-rΔ) is an ideal sheaf, so O_X(-⌈nΔ⌉-rΔ)⊗E_n^{-1} is a subsheaf of O_X(-⌈nΔ⌉)⊗E_n^{-1}, not a super-sheaf. The h^0 inequality therefore has the wrong direction, and the displayed contradiction does not follow. As a result, Proposition 4.7, and with it Theorem 4.2 and Corollary 4.3, are not proved as written. The gap appears repairable: in the remaining case deg(A) ≤ 2p_a(C)-2, so h^0(A) is bounded, while Eq. (19) and the linear growth of h^0(L_C) force h^0(A) to grow linearly; this contradiction would eliminate the remaining case for n ≫ 0, but this argument is not supplied in the manuscript.","section":"Section 4, Proposition 4.7"},{"comment":"Theorems A and B and their corollaries rely on Theorem 1.1, which is quoted from the authors' unpublished companion preprint [20] (arXiv:2503.14429), and Theorem D depends on the structural representation of Q-divisorial filtrations from [15]. The present manuscript is therefore conditional on the correctness of those external inputs. The dependence is stated, but the authors should make explicit that the main results cannot be verified independently of [20] and should indicate whether [20] is under review or available in final form.","section":"Sections 1 and 3"}],"minor_comments":[{"comment":"In the proof of Proposition 2.2, the sentence 'f_i h = 0 if and only if Y is not in the support of Div(σ)' is backwards; the intended statement is that the restriction map is injective if and only if Y is not in the support of Div(σ). The proposition itself is correct, but the explanatory sentence should be fixed.","section":"Section 2, Proposition 2.2"},{"comment":"The displayed expression 'v_0(x)+v_0(x)+v_1(x)+···+v_n(x)/n' is missing parentheses; it should read ((n+1)v_0(x)+∑_{i=1}^n v_i(x))/n, or equivalently v_0(x)+(v_0(x)+···+v_n(x))/n.","section":"Example 4.1, before Eq. (10)"},{"comment":"In the line '∑_{v∈Div(nR)} v(x)lim...' the notation 'Div(nR)' should be 'Div(m_R)'.","section":"Example 4.1, after Eq. (9)"},{"comment":"Reference [48] lists the year as 1050; this should be 1950.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The sign error in Proposition 4.7 is a genuine gap in the proof of Theorem D, but it is localized and the alternative boundedness argument sketched in my report would likely repair it. I therefore recommend major revision rather than rejection. The heavier concern for the editor may be the paper's reliance on the companion preprint [20] for the foundational intersection-theoretic machinery; if the journal permits citation of preprints, this is acceptable, but it should be made very explicit in the final version."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a solid extension of the authors' intersection-product framework to graded families, and the dimension-2 examples are genuinely new. But the proof of Theorem D does not close as written. The stress-test note is correct: in the remaining case of Proposition 4.7, with A=O_X(-ceil nΔ)⊗E_n^{-1} and B=A⊗O_X(-rΔ) on C, the sheaf O_X(-rΔ) is an ideal sheaf because rΔ is effective, so B is a subsheaf of A and h^0(B) ≤ h^0(A). The text asserts the opposite inequality. The chain needs h^0(A) ≤ h^0(B)=χ(B), and with the actual inclusion the inequality points the wrong way. So the contradiction e>...>e is not established. There may be a way to repair the remaining case, e.g. a degree-growth argument, but it is not supplied. This affects Theorem 4.2 and Corollary 4.3, which are exactly the new closed formula in dimension 2.\n\nThe rest of the paper is in decent shape. Theorem A and Theorem B follow cleanly from the multiplicity formula and Theorem 3.2; I do not see an internal error there. Corollary C is a straightforward interchange argument under the finiteness and existence hypotheses. Example 4.1 is a nice construction showing noncommutation of limit and sum, and Examples 5.2 and 5.3 give the promised finite and infinite Rees valuation sets for divisorial filtrations. Those are real contributions. The proof of Example 4.1 uses Lemma 4.6 of the unpublished preprint [5], and Theorems A and B inherit [20], so independent verification of those inputs is still needed. That is a conditionality, not by itself a flaw in the present arguments.\n\nBottom line: the paper deserves a serious referee, but not acceptance in current form. I would send it to review with a clear request to fix Proposition 4.7 and to state the dependence on [20] and [5]. If the gap is repaired, Theorem D becomes a useful result; the examples alone justify circulation.","headline":"Useful paper with a real gap: Theorem D's key inequality is reversed in Proposition 4.7; the earlier theorems and new examples are still worth referee time.","tokens_in":26568,"tokens_out":2618,"would_cite":false,"duration_ms":30054,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["13H15","14C17","13A30","13B22","14E15"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that Rees-style degree functions extend from single ideals to graded families of m-primary ideals, expressing multiplicities as limits of intersection products and giving closed formulas in dimension 2.","keywords":["graded families of ideals","divisorial filtrations","Rees valuations","degree functions","multiplicities","intersection products","excellent local rings","Hilbert-Samuel multiplicity"],"falsifier":"A concrete falsifier would be an explicit Q-divisorial filtration on a 2-dimensional excellent normal local ring together with an m-valuation v for which the limit of (I_nO_{Y_n}·C(Y_n,v))/n is not equal to (-$\\Delta$·C(X,v)), despite $\\Delta$ satisfying the representation property; or a graded family for which the limit in Theorem B fails to exist.","tokens_in":25577,"feed_emoji":"🧮","tokens_out":4462,"duration_ms":52785,"temperature":0.7,"pith_summary":"This paper extends Rees's classical degree function, defined for a single m-primary ideal, to graded families of m-primary ideals in excellent normal local rings. It proves that the multiplicity of the family modulo a reduced principal ideal is a limit of sums of intersection products over centers of m-valuations, and that under a finiteness condition the limit and the sum commute to give a genuine degree function. In dimension 2, for Q-divisorial filtrations, the per-valuation coefficients stabilize and are computed by intersecting an effective exceptional Q-divisor with the valuation's center. The paper also constructs examples showing that the union of all Rees valuations of a graded family can be finite or infinite, and that limits and sums need not commute in general.","feed_headline":"Rees degree function now covers graded families of ideals","feed_subtitle":"In dimension 2, divisorial filtrations get closed-form coefficients from a single Q-divisor.","key_machinery":"The load-bearing object is the intersection product (-)_R defined on schemes proper and birational over a local ring, together with the center C(Y_n,v) of an m-valuation on the normal model where I_nO_{Y_n} is invertible. For dimension 2, the filtration is represented by a Q-divisor $\\Delta$ with exceptional support such that Gamma(X,O_X(-ceil(nDelta)))=I_n and -$\\Delta$ is nef; the coefficient (-$\\Delta$·C(X,v)) is the limit that appears in the degree formula.","core_discovery":"The central discovery is that degree functions, classically attached to a single m-primary ideal via its Rees valuations, make sense for entire graded families after passing to limits: for an excellent normal local ring, e(I(R/xR)) equals the limit as n tends to infinity of (1/$n^{{d-1}}$) times the sum over m-valuations v of v(x)((I_nO_{Y_n})^{d-1}·C(Y_n,v))_R. In dimension 2, for Q-divisorial filtrations, this simplifies to a linear combination with coefficients (-$\\Delta$·C(X,v)) that exist and equal the limit of the normalized per-valuation intersection numbers. When the union of the Rees valuations of all members is finite, the resulting degree function is a finite sum of valuation terms weighted by these intersection numbers.","pith_inferences":["The limit formulas suggest that a graded family of ideals carries an asymptotic degree function that can be thought of as a limit of the degree functions of its members; one might expect convex-geometric analogues via Newton-Okounkov bodies in higher dimensions.","The question of when the per-valuation limits exist (Question 1.3) is central; the dimension-2 answer hints that such limits may fail in higher dimensions or without the Q-divisorial representation hypothesis.","The examples with zero analytic spread show that finiteness of the total set of Rees valuations is not a Noetherian property but rather a boundedness condition on normalized intersection numbers; one could test whether it is equivalent to uniform boundedness of these numbers.","Since the paper relies on an unpublished companion preprint for the intersection product, the sharpest check of the results would be a direct computation of the formulas on the explicit elliptic-curve examples given here."],"forward_implications":["Multiplicities of graded families of m-primary ideals modulo reduced principal ideals are computable as limits of intersection products, generalizing Rees's classical degree function.","The limit and the sum in the degree formula commute whenever the union of the Rees valuations of all members is finite and each per-valuation limit exists, giving a finitely supported degree function.","For Q-divisorial filtrations on 2-dimensional excellent normal local rings, the per-valuation limits always exist and equal (-Delta·C(X,v)), so the degree function has an explicit closed form.","Noetherian filtrations always have finitely many Rees valuations in total, so the finite-union condition is automatically satisfied in the Noetherian case.","There exist non-Noetherian divisorial filtrations whose union of Rees valuations is still finite, and others for which that set is infinite, showing that the finiteness condition is independent of Noetherianity."],"supporting_citations":[{"why":"Develops the intersection product (-)_R and proves the geometric formulas for multiplicities and degree functions of a single ideal that the graded-family theorems extend.","marker":"[20]"},{"why":"Establishes the existence of the asymptotic multiplicity e(I) for graded families and provides the volume=multiplicity formula used in the proof of Theorem A.","marker":"[12]"},{"why":"Supplies the structural representation of Q-divisorial filtrations in dimension 2 by an effective Q-divisor Delta with -Delta nef and Gamma(X,O_X(-ceil(nDelta)))=I_n, on which Theorem D depends.","marker":"[15]"},{"why":"Contains Rees's original degree function theorem that this paper generalizes to graded families.","marker":"[45]"},{"why":"Provides the background facts on Rees valuations and divisorial valuations used to identify the canonical valuations with Rees valuations.","marker":"[51]"},{"why":"Used to conclude that R/xR is analytically unramified when R is excellent and R/xR is reduced, a key step in the proof of Theorem B.","marker":"[26]"}],"fun_headline_variants":["Degree functions via limits for graded ideals","Intersection limits define degree for ideals","Q-divisors simplify degree functions in dim 2","Non-Noetherian filtrations: finite or infinite valuations"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The dimension-2 formula assumes, via the cited structural result, that every Q-divisorial filtration of m-primary ideals on a 2-dimensional excellent normal local ring can be represented as Gamma(X,O_X(-ceil(nDelta))) with $\\Delta$ effective, exceptional, and -$\\Delta$ nef, and the general theorems assume the correctness of the companion preprint that constructs the intersection product.","fun_headline_variants_meta":{"raw":{"variants":["Degree functions via limits for graded ideals","Intersection limits define degree for ideals","Q-divisors simplify degree functions in dim 2","Non-Noetherian filtrations: finite or infinite valuations"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000188,"raw_usage":{"total_tokens":1274,"prompt_tokens":830,"completion_tokens":444,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":446,"completion_tokens_details":{"reasoning_tokens":385}},"tokens_in":446,"tokens_out":444,"duration_ms":5210,"temperature":1.0,"reasoning_tokens":385,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T10:22:33.246517+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete falsifier would be an explicit Q-divisorial filtration on a 2-dimensional excellent normal local ring together with an m-valuation v for which the limit of (I_nO_{Y_n}·C(Y_n,v))/n is not equal to (-$\\Delta$·C(X,v)), despite $\\Delta$ satisfying the representation property; or a graded family for which the limit in Theorem B fails to exist.","supporting_citations":[{"cited_title":"Multiplicities and degree functions in local rings via intersection products","cited_arxiv_id":"2503.14429","evidence_quote":"Develops the intersection product (-)_R and proves the geometric formulas for multiplicities and degree functions of a single ideal that the graded-family theorems extend."},{"cited_title":"Cutkosky, Asymptotic multiplicities of graded families of ideals and linear series, Adv","cited_arxiv_id":null,"evidence_quote":"Establishes the existence of the asymptotic multiplicity e(I) for graded families and provides the volume=multiplicity formula used in the proof of Theorem A."},{"cited_title":"Cutkosky, Analytic Spread of filtrations on two dimensional normal local rings, Nagoya Math","cited_arxiv_id":null,"evidence_quote":"Supplies the structural representation of Q-divisorial filtrations in dimension 2 by an effective Q-divisor Delta with -Delta nef and Gamma(X,O_X(-ceil(nDelta)))=I_n, on which Theorem D depends."},{"cited_title":"Rees, Degree functions in local rings, Proceedings of the Cambridge Phil","cited_arxiv_id":null,"evidence_quote":"Contains Rees's original degree function theorem that this paper generalizes to graded families."},{"cited_title":"Swanson and C","cited_arxiv_id":null,"evidence_quote":"Provides the background facts on Rees valuations and divisorial valuations used to identify the canonical valuations with Rees valuations."},{"cited_title":"Grothendieck, Éléments de géométrie algébrique IV , Étude locale des Schémas et des Morphismes des Schémas (Seconde Partie), Pub","cited_arxiv_id":null,"evidence_quote":"Used to conclude that R/xR is analytically unramified when R is excellent and R/xR is reduced, a key step in the proof of Theorem B."}],"review_version":1}