{"id":"5c872bb8-a528-4eee-89be-097436b45e0c","arxiv_id":"1908.02474","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For toric plurisubharmonic functions in dimension 2, the cluster points of jumping numbers are exactly the positive integer multiples of 1/x0 and 1/y0, determined by the asymptotes of the Newton convex body.","lead":"This math paper studies 'jumping numbers', the special coefficients where multiplier ideal sheaves attached to a plurisubharmonic function suddenly shrink. The authors show that in the two-variable toric case, these jumps accumulate exactly when a coordinate asymptote of the Newton convex body lies outside it, and they list all possible accumulation points.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 5.7's 'only if' direction depends on an unproved subsequence extraction: a cluster point is converted into an asymptote line only after asserting, without proof, that the sequence A_k has a subsequence with one coordinate constant and the other strictly increasing.","rationale":"The paper's main theorem is interesting and, in my reading, very likely correct: the convex-geometric mechanism of the theorem is coherent, and the two known examples fit it precisely. The external theorem singled out by the reader, Proposition 2.7 of Guenancia–Rashkovskii, is a published result and is not the weakest point of the paper's own argument. The genuinely load-bearing gap is internal: the proof of Theorem 5.7 asserts a subsequence with one coordinate constant and the other strictly increasing without proving that a bounded coordinate exists. Since this step is what turns a cluster point into a statement that c x0 or c y0 is an integer, the necessity direction of the characterization is incomplete as printed. The gap is likely repairable via an upward-closure argument, and the same repair would also handle the missing closure step in Theorem 5.3 and Proposition 3.3. The appendix by Boucksom appears to use standard valuative criteria and, while terse, is not where the central claim is at risk. Therefore I do not recommend changing the reader's CONDITIONAL verdict; I recommend keeping it conditional pending a written bounded-coordinate argument for the subsequence extraction, rather than rejecting the paper.","tokens_in":18997,"tokens_out":33262,"duration_ms":358182,"concrete_test":"Settle the subsequence claim by an analytical check plus a concrete model: (a) prove the bounded-coordinate lemma that any sequence q_k = A_k+1 satisfying the two inclusions in the proof of Theorem 5.7 has one coordinate eventually bounded, because otherwise q_k would lie in intP(cϕ) for large k; (b) for the concrete upward-closed Newton body P = {(x−√2)y ≥ 1} (x0 = √2 > 0, vertical asymptote disjoint, y0 = 0), write the slice inequalities explicitly for c = k/√2 and ε_k = 1/k, and test whether integer sequences with both coordinates diverging can satisfy the two inclusions. If such a sequence exists, the asserted constant-coordinate subsequence is false and the necessity direction of Theorem 5.7 is not valid as stated; if none exists, the missing bounded-coordinate lemma should be written into the proof.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In the necessity half of Theorem 5.7 (proof, §5.2), the authors produce integer vectors A_k with A_k+1 ∈ intP((c−ε_k)ϕ) and A_k+1 ∉ intP((c−ε_{k+1})ϕ), where ε_k ↓ 0. From the second condition and upward closure of P(cϕ), exactly one coordinate of A_k increases in each step. The proof then states: 'One can find a subsequence (A_{k_j}) such that either pr1 is constant and pr2 strictly increases, or vice versa.' This is not a formal consequence of the preceding sentence: a sequence could alternate which coordinate increases or have one coordinate strictly increasing while the other strictly decreases, with no coordinate ever constant. The inference from a cluster point to 'cx0 is an integer' depends entirely on this extraction, so this is the load-bearing step of the main characterization. The gap appears repairable: since A_k+1 ∉ intP(cϕ), a coordinate that eventually exceeds the corresponding asymptote (c x0 or c y0) cannot diverge; hence at least one coordinate is bounded and a constant subsequence exists, after which the other coordinate must tend to infinity. But the printed proof omits this bounded-coordinate argument. The analogous missing step also appears in the proof of Theorem 5.3, where a jumping number c is reduced to the case c ∈ C(ϕ) without explicitly handling c ∈ Jump(ϕ) \\ C(ϕ) through the closure statement of Proposition 3.3. These are proof gaps, not necessarily false claims, but as written the central theorem is not fully established.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper extends the study of jumping numbers of multiplier ideals from the algebraic setting (Ein–Lazarsfeld–Smith–Varolin) to general plurisubharmonic functions. After establishing basic properties via the Guan–Zhou openness theorem, the authors analyze v-equivalence of psh functions, construct Siu psh functions from graded systems of ideals, and prove several examples. The main result, Theorem 5.7, gives a complete characterization, in terms of the Newton convex body, of when a toric psh function on the unit polydisk in C^2 has cluster points of jumping numbers at the origin, and identifies the set of such cluster points. An appendix by Boucksom proves that analytic multiplier ideals of Siu psh functions coincide with the asymptotic multiplier ideals of the underlying graded system.","tokens_in":19200,"tokens_out":7951,"duration_ms":79671,"significance":"If the proof gaps are repaired, Theorem 5.7 would be a substantial contribution: it completely describes the cluster-point phenomenon for toric psh functions in dimension two, generalizing the two known examples of Guan–Li and Ein–Lazarsfeld–Smith–Varolin and producing uncountably many new examples. The convex-geometric formulation is elegant, and the appendix by Boucksom is of independent interest, as it settles a natural comparison between analytic and algebraic asymptotic multiplier ideals. The paper is clearly written and the main statement is plausible and consistent with the worked examples.","major_comments":[{"comment":"The sequence conditions (1)–(2) in the proof are internally inconsistent. Since ε_{k+1} < ε_k, the set intP((c−ε_{k+1})ϕ) is a superset of intP((c−ε_k)ϕ), so no lattice point A_k+1 can satisfy both A_k+1 ∈ intP((c−ε_k)ϕ) and A_k+1 ∉ intP((c−ε_{k+1})ϕ). The same inconsistency appears in the sufficiency direction with B_j. The intended condition presumably involves intP(cϕ) or intP((c+ε_{k+1})ϕ); as written, the existence of the sequence is impossible, so the proof of the cluster-point characterization is not valid.","section":"§5.2, proof of Theorem 5.7"},{"comment":"The subsequence extraction is unjustified and, as stated, false. From 'exactly one coordinate of A_k increases at each step' it does not follow that there is a subsequence with one coordinate constant and the other strictly increasing; for example, the sequence (0,0),(0,1),(1,1),(1,2),(2,2),(2,3),... has both coordinates unbounded along every subsequence. The proof needs an additional argument showing that because A_k+1 ∉ intP(cϕ), at least one coordinate is bounded along a subsequence; this is exactly the step that converts a cluster point into a vertical or horizontal asymptote, so the main theorem is not established as written.","section":"§5.2, proof of Theorem 5.7"},{"comment":"The proof only treats the case c ∈ C(ϕ). For a jumping number c that lies in the closure of C(ϕ) but not in C(ϕ), Proposition 3.3 gives c ∈ closure(C(ϕ)), but the argument showing mC(ϕ) ⊂ C(ϕ) does not directly imply mc ∈ Jump(ϕ)_0. One needs an approximation argument using the fact that Jump(ϕ)_0 is closed, or an alternative argument; as printed, the theorem is not proved for cluster-point jumping numbers.","section":"§5.1, proof of Theorem 5.3"}],"minor_comments":[{"comment":"There are several typographical errors: 'absense' in the introduction, 'Newon' in Example 3.12, and 'Theorem 5.10' in the proof of Corollary 5.11 should likely be Theorem 5.7 or 5.3.","section":"Throughout"},{"comment":"The proof of the inclusion for non-cluster jumping numbers assumes that a single lattice point A can be chosen independently of ε; this deserves a brief justification using Noetherianity of the stalks.","section":"§3.1, Proposition 3.3"},{"comment":"The construction of the sequence (α_j) in the sufficiency direction is only sketched; it would be helpful to explain explicitly why such lattice points can be chosen with the stated nesting property, especially given the inconsistency noted in the first major comment.","section":"§5.2, sufficiency direction"},{"comment":"The proof of Theorem 2.2 is very terse; in particular, the passage from v(ϕ) = v(a•) to the multiplier ideal equality uses the valuative criteria without recalling the normalization condition on v in the singular case. This is understandable for an appendix by an expert but could be clarified for the reader.","section":"Appendix"}],"recommendation":"major_revision","confidential_remarks":"The main theorem is plausible and the paper contains interesting ideas, but the proof gaps in §5 are load-bearing. I would be willing to look at a revised version. The appendix by Boucksom is a strong independent contribution."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know this paper is worth a serious look. The main result, Theorem 5.7, gives a complete convex-geometric criterion for when a toric psh function on the unit polydisk in C^2 has cluster points of jumping numbers, and it identifies all cluster points as the positive rationals k/x0 and/or k/y0. That is a real advance: previously only two sporadic examples (Guan–Li and ELSV) were known, and this paper shows clustering is a frequent phenomenon and answers Question 1.1 affirmatively in this toric setting. The appendix by Boucksom, proving that analytic multiplier ideals of Siu psh functions equal asymptotic multiplier ideals, is an independent and cleanly proved result that will be cited on its own.\n\nThe paper does several things well. The reduction to Newton convex bodies is natural, and Proposition 3.3, linking jumping numbers to lattice points on the boundary of P(cϕ), is a useful tool. Corollary 5.11 sharpening the Guan–Li example, and the explicit computations in Examples 3.6 and 3.12, strengthen the overall picture. The writing is clear and the authors are honest about what depends on external theorems like Guenancia–Rashkovskii and the openness conjecture.\n\nThat said, the proof of the central theorem, as printed, has some genuine gaps. In the proof of Theorem 5.7, condition (2) says A_k+1 ∈ intP((c−ε_k)ϕ) but A_k+1 ∉ intP((c−ε_{k+1})ϕ). Since ε_k > ε_{k+1}, the second set is larger, so that containment pattern is impossible. The later text and the intended argument use “not in intP(cϕ)”, so this looks like a simple typo, but it has to be fixed. More substantively, the proof asserts without argument that a subsequence exists with one coordinate constant and the other strictly increasing. That extraction is not a formal consequence of the preceding line; it needs the bounded-coordinate argument the stress-test note sketches. This is the load-bearing step in the necessity direction, so it cannot be left as is. Theorem 5.3 has a smaller version of the same issue: it only proves mC(ϕ) ⊂ C(ϕ), then jumps to all jumping numbers without explicitly using the closure statement of Proposition 3.3 for c ∈ Jump(ϕ) \\ C(ϕ). All these look repairable, and I do not see a reason to doubt the main theorem, but as printed the proof is incomplete.\n\nWho is this for? Anyone working on pluripotential theory, multiplier ideals, or singularities of psh functions will get something from it. The gaps are annoying but not fatal. I would recommend that the editors send it to a serious referee who can ask for a repaired proof. My own default would be to accept it after that repair.\n\nI would bring it to a reading group and would cite the Boucksom appendix in my own work.","headline":"A genuinely new complete characterization of cluster points of jumping numbers for toric psh functions in dimension 2, with repairable gaps in the proof and a valuable appendix by Boucksom.","tokens_in":19883,"tokens_out":2858,"would_cite":true,"duration_ms":29800,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14B05","14F18","32U05","32U25"],"pacs":[],"model":"deepseek-v4-flash","headline":"For toric singularities on the unit bidisk, jumping numbers have cluster points exactly when the Newton convex body fails to meet one of its coordinate asymptotes, and then all cluster points are explicit rational numbers.","keywords":["jumping numbers","multiplier ideals","plurisubharmonic functions","toric psh functions","Newton convex body","cluster points","asymptotic multiplier ideals","periodicity"],"falsifier":"Take a toric psh function on $D^2$ whose Newton convex body $P$ has a horizontal asymptote $y=y_0>0$ disjoint from $P$ and a vertical asymptote at $x=0$ meeting $P$; Theorem 5.7 predicts exactly the cluster points $\\{k/y_0:k\\in\\mathbb{Z}_{>0}\\}$. Compute $\\mathrm{Jump}(\\phi)_0$ directly as the closure of $C(\\phi)$ from lattice points on $\\partial P(c\\phi)$; any cluster point outside that predicted set, or any missing predicted point, would falsify the characterization.","tokens_in":18643,"feed_emoji":"📐","tokens_out":11305,"duration_ms":97815,"temperature":0.7,"pith_summary":"This paper extends the theory of jumping numbers of multiplier ideals from algebraic singularities to general plurisubharmonic (psh) functions. Many algebraic properties survive, but two central ones do not: jumping numbers need not be discrete and need not be periodic. For toric psh functions on the unit bidisk, the authors fully characterize when jumping numbers accumulate: clustering happens exactly when one of the coordinate asymptotes of the Newton convex body does not meet the body. In that case the cluster points are precisely the rational numbers $k/m$ formed from the asymptote's distance to the origin, and this yields uncountably many new examples with clustering behavior.","feed_headline":"Jumping numbers cluster exactly when a convex body misses an asymptote","feed_subtitle":"For toric singularities in two variables, cluster points are the reciprocals of the missing asymptote scaled by integers.","key_machinery":"The load-bearing object is the Newton convex body $P(\\phi)$ of a toric psh function, together with the criterion (Proposition 2.7) that the multiplier ideal $J(\\phi)$ is monomial and $z_1^{a_1}\\cdots z_n^{a_n}\\in J(\\phi)$ if and only if $(a_1+1,\\dots,a_n+1)$ lies in the interior of $P(\\phi)$. This turns the analytic question of when multiplier ideals jump into the convex-geometric question of which lattice points cross the boundary $\\partial P(c\\phi)$ as $c$ varies. In dimension 2, the asymptotic behavior of the boundary of $P(\\phi)$ near its two coordinate asymptotes then determines all cluster points.","core_discovery":"For a toric psh function $\\phi$ on $D^2$ with Newton convex body $P(\\phi)$, write $x_0=\\inf \\mathrm{pr}_1(P(\\phi))$ and $y_0=\\inf \\mathrm{pr}_2(P(\\phi))$. Theorem 5.7 states that $\\mathrm{Jump}(\\phi)_0$ has a cluster point if and only if either $x_0>0$ and the vertical line $\\{(x_0,t):t\\in\\mathbb{R}\\}$ misses $P(\\phi)$, or $y_0>0$ and the horizontal line $\\{(t,y_0):t\\in\\mathbb{R}\\}$ misses $P(\\phi)$. In that case the set of cluster points is exactly $\\{k/m:k\\in\\mathbb{Z}_{>0},\\, m\\in S\\}$, where $S\\subseteq\\{x_0,y_0\\}$ contains $x_0$ exactly in the first case and $y_0$ exactly in the second. The paper also proves that for toric psh functions on $D^n$, if $c$ is a jumping number then all integer multiples $mc$ are jumping numbers, and that periodicity of jumping numbers fails in general; the appendix proves that asymptotic multiplier ideals of a graded system of ideals equal the analytic multiplier ideals of the associated psh function.","pith_inferences":["Because the asymptote-disjoint condition is open in the space of Newton convex bodies, cluster-point behavior should be generic among toric psh singularities in dimension 2, not a rare pathology.","The same asymptote criterion can be tested for mixed jumping numbers $\\mathrm{Jump}(\\phi;\\psi)_0$ by Minkowski addition of the two Newton bodies, giving an explicit recipe for clustering in additive families.","In higher dimensions the natural extension is that cluster points are governed by coordinate hyperplane asymptotes of $P(\\phi)$; the dimension-2 proof suggests the cluster set will be generated by reciprocals of the missing asymptote distances, possibly with new phenomena such as cluster points of cluster points.","The equality between asymptotic and analytic multiplier ideals suggests one can engineer psh singularities with prescribed cluster sets by choosing Newton convex bodies with the desired asymptote structure, which may inform the open algebraic question of whether cluster points occur for multiplier ideals on singular varieties."],"forward_implications":["If a toric psh function in dimension 2 has one cluster point, then it has infinitely many cluster points, and they form an unbounded set.","Every cluster point of jumping numbers is itself a jumping number, so the failure of discreteness occurs at values that still belong to the jumping set.","For the earlier psh example whose jumping numbers accumulate at 1, the full set of cluster points is precisely the positive integers.","Periodicity of jumping numbers, which holds in the algebraic case, can fail for general psh functions; an explicit example with no period is presented.","The asymptotic multiplier ideal of a graded system of ideals equals the analytic multiplier ideal of the associated psh function, so algebraic and analytic constructions can be used interchangeably in this setting."],"supporting_citations":[{"why":"Openness theorem that makes jumping numbers well-defined for general psh functions by guaranteeing constancy on half-open intervals.","marker":"[GZ]"},{"why":"Provides the monomial criterion for multiplier ideals of toric psh functions, the bridge to Newton convex bodies.","marker":"[G]"},{"why":"Supplies the Newton convex body construction and the monomial membership criterion used throughout.","marker":"[R13b]"},{"why":"Foundational algebraic theory of jumping coefficients and the graded-system example whose cluster points are all positive integers.","marker":"[ELSV]"},{"why":"First psh example with a cluster point; Corollary 5.11 refines it by identifying all cluster points.","marker":"[GL]"},{"why":"Valuative characterization of multiplier ideals used in the appendix to prove equality with asymptotic multiplier ideals.","marker":"[BFJ]"},{"why":"Valuative characterization of asymptotic multiplier ideals used in the appendix to complete the equality proof.","marker":"[BFFU]"},{"why":"Provides the valuation formula for graded systems that identifies the valuations of the associated psh function.","marker":"[JM]"},{"why":"Provides the example of a psh function whose jumping numbers fail periodicity, analyzed in Section 4.","marker":"[Ko15]"}],"fun_headline_variants":["Jumping numbers cluster iff a line misses the Newton body","Cluster points of jumping numbers occur when a line misses the Newton body","Toric psh functions: jumping numbers cluster iff an axis line is missing","When do jumping numbers cluster? When a Newton body misses an axis line","Toric psh jumping numbers: cluster points from missing lines"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The central argument assumes that a toric psh function's multiplier ideal is completely encoded by the integer lattice points inside its Newton convex body, via the criterion that $z^A$ belongs exactly when $A+\\mathbf{1}$ lies in the interior; if that encoding failed for even one toric psh function, the lattice-point boundary analysis behind the cluster-point theorem would collapse.","fun_headline_variants_meta":{"raw":{"variants":["Jumping numbers cluster iff a line misses the Newton body","Cluster points of jumping numbers occur when a line misses the Newton body","Toric psh functions: jumping numbers cluster iff an axis line is missing","When do jumping numbers cluster? When a Newton body misses an axis line","Toric psh jumping numbers: cluster points from missing lines"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001028,"raw_usage":{"total_tokens":4350,"prompt_tokens":981,"completion_tokens":3369,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":597,"completion_tokens_details":{"reasoning_tokens":3278}},"tokens_in":597,"tokens_out":3369,"duration_ms":23445,"temperature":1.0,"reasoning_tokens":3278,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:44:46.018953+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a toric psh function on $D^2$ whose Newton convex body $P$ has a horizontal asymptote $y=y_0>0$ disjoint from $P$ and a vertical asymptote at $x=0$ meeting $P$; Theorem 5.7 predicts exactly the cluster points $\\{k/y_0:k\\in\\mathbb{Z}_{>0}\\}$. Compute $\\mathrm{Jump}(\\phi)_0$ directly as the closure of $C(\\phi)$ from lattice points on $\\partial P(c\\phi)$; any cluster point outside that predicted set, or any missing predicted point, would falsify the characterization.","supporting_citations":[],"review_version":1}