{"id":"d811ac43-ef35-4742-a351-b41eefb5893c","arxiv_id":"2411.09107","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"high","formal_verification":"none","parameter_count":1,"one_line_summary":"Using Bridgeland stability on normal surfaces, the authors prove Reider-type separation-of-jets bounds for ω_X⊗L^a, including positive characteristic and Du Bois variants, recovering optimal Fujita constants when C_X=0.","lead":"A new proof of Reider-type vanishing theorems for adjoint bundles on normal surfaces using Bridgeland stability, extending Arcara-Bertram to singular and positive characteristic settings. It also treats a Du Bois variant of the canonical sheaf.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4.7 is not proved for partitions with l1 > l2: the stated H² hypothesis is too weak to define the stability condition t1 used in the proof.","rationale":"After good-faith reading, the central construction is coherent: Langer's Bridgeland stability, the type O formalism, and the reduction of the H¹-vanishing to Hom(L⊗I_Z,F)=0 are all consistent with the cited framework. The reader's flagged degree inconsistency in Proposition 3.5 appears to be a typesetting artifact: if d_{s,t}(E) is read as ch2(E) − s ch1(E)·H + r((s²−t²)H² − C_X)/2, then the substitution t² = 1/4 − (2l+C_X)/H² yields exactly the claimed bound A·H ≤ A²/r + r(C_X+2l). The genuine soft spot is the l1 > l2 case of Theorem 4.7. The proof needs H² > 4(2l1+C_X) for Proposition 3.9, but the theorem only supplies H² ≥ 4(l1+l2+C_X)+ε; for l1 > l2 this is strictly weaker, so the t1 stability condition is not defined. This is an internal gap in the central technical statement, not merely a reliance on an external constant. The advertised Fujita bounds use balanced or minimizing partitions with l1≤l2, so the main applications may survive a corrected theorem; hence the verdict should remain conditional rather than move to reject. The concrete test isolates the minimal unbalanced case and would determine whether the theorem is false or just under-proved as written.","tokens_in":20699,"tokens_out":42081,"duration_ms":440348,"concrete_test":"Work through the minimal unbalanced case l1=1, l2=0, C_X=0. For an ample H with 5 ≤ H² < 8 satisfying the theorem's H·C ≥ 4 for every effective curve, compute t1² = 1/4 − 2/H²: it is negative, so Proposition 3.9 cannot be invoked at t1. Then decide independently, by Reider's theorem or by a direct search over smooth polarized surfaces, whether H¹(ω_X⊗H⊗I_p)=0 for every such H and every closed point p. If a nonzero H¹ is found, Theorem 4.7 is false as stated; if no such case exists, the conclusion may still hold, but the written proof must be repaired. Either outcome settles whether the gap is merely a proof issue or an error in the theorem's hypotheses.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 4.7 reduces to Hom(L⊗I_{Z'},F') with lZ'=l1 and lF'=l2, then uses Proposition 3.11 to make Im f a torsion sheaf G. To rule out G as a destabilizing quotient of L⊗I_{Z'}, it invokes Proposition 3.9 at the stability condition (s=1/2, t1), where t1² = 1/4 − (2l1+C_X)/H². This requires H² > 4(2l1+C_X). The theorem's hypotheses give only H² ≥ max((C_X+2l2+1)², 4(l1+l2+C_X)+ε). When l1 > l2, we have l1+l2 < 2l1, so the second term is strictly smaller than 4(2l1+C_X)+ε and the first term can also be smaller. Thus t1 is not real and Proposition 3.9 cannot be applied. The subsequent transfer of d(G)>0 from t1 to t2 collapses. For example, with l1=2, l2=0, and C_X=0, the proof requires H² > 16, while the stated bound is only H² ≥ 8+ε. The theorem as written is therefore unproved for every unbalanced partition. The Fujita applications are not directly affected because the minimizers in m(C_X,l) have l1≤l2, but Theorem 4.7 is the technical core and its stated generality is not supported. The repair is either to restrict Theorem 4.7 to l1≤l2, or to replace the H² hypothesis by 4(2l1+C_X)+ε when l1>l2, or to provide a new argument avoiding the t1 stability condition.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper proves Reider-type vanishing and jet-separation theorems on projective normal surfaces over algebraically closed fields, using Langer's Bridgeland stability conditions on normal surfaces. The main technical result (Theorem 4.7) gives the vanishing of Hom(L⊗I_Z,F) for an ample line bundle L, a zero-dimensional subscheme Z, and an object F of type O (with cohomology matching O_X[1] or Ω^0_X[1]), under lower bounds on H²=c1(L)² and on H·C involving a surface-dependent constant C_X. From this the authors deduce H¹(X,ω_X⊗L⊗I_Z)=0, separation of jets, and Fujita-type bounds: for C_X=0 (e.g., complex surfaces with at worst rational double points) ω_X⊗L^a is globally generated for a≥3 and very ample for a≥4, recovering the optimal Fujita constants. In positive characteristic, explicit C_X from Koseki's Bogomolov inequality yield corresponding (weaker) bounds, and for complex surfaces the same Reider machinery is applied to the Du Bois dual ω_DB = RHom(Ω^0_X,ω_X).","tokens_in":21075,"tokens_out":22170,"duration_ms":165572,"significance":"If the technical issues flagged below are repaired, this is a substantial and welcome extension of the Arcara–Bertram/Bridgeland approach to Reider-type theorems, covering normal surfaces, positive characteristic, and a Du Bois variant. The paper makes essential, clearly attributed use of Langer's stability conditions and Bogomolov-type inequalities; the geometric constant C_X is explicit and computed in examples. The optimal Fujita constants 3 and 4 are recovered exactly when C_X=0. The main results are parameter-free in the sense that the only geometric input is C_X, and the paper explicitly identifies where this constant appears. The manuscript is well organized and the strategy of reducing to torsion images is appealing.","major_comments":[{"comment":"Under the stated formula t² = 1/4 − (2l+C_X)/H², the proof of Proposition 3.5 is inconsistent with the conclusion. With s=1/2, s²−t² = (2l+C_X)/H², and the degree satisfies d_{s,t}(E) = ch2(E) − (1/2)A·H + r(2l+C_X) − (C_X/2)r, so the inequality ν ≤ 0 with r ≤ 0 yields A·H ≤ 2ch2(E) + 4rl + C_X r, not A·H ≤ 2ch2(E) + 2rl. The claimed bound A·H ≤ A²/r + r(C_X+2l) is only consistent with t² = 1/4 − (2l+C_X)/(2H²). This is load-bearing because the constants in the Reider bounds and the realness condition for t are affected. The same typo appears in Propositions 3.7, 3.8, 3.9, 3.10, and Theorem 4.7; the authors should correct the formula in all these statements (and in the proof of Proposition 3.9, whose line '+l_Z' already uses the corrected form).","section":"§3, Proposition 3.5 and Definition of t"},{"comment":"As written, Theorem 4.7 is not proved for partitions with l1 > l2. The proof applies Proposition 3.9 at the stability condition with t1² = 1/4 − (2l1+C_X)/H², which requires H² > 4(2l1+C_X). The theorem's hypothesis only guarantees H² ≥ max((C_X+2l2+1)², 4(l1+l2+C_X)+ε). When l1 > l2, the second term is smaller than 4(2l1+C_X) (for example, l1=2, l2=0, C_X=0 gives a required H² > 16 while the hypothesis gives H² ≥ 8+ε), and the first term may also be smaller, so t1 is not real and Proposition 3.9 cannot be invoked. If the t² formula is corrected to t² = 1/4 − (2l1+C_X)/(2H²), then the hypothesis H² ≥ 4(l1+l2+C_X)+ε implies H² > 4l1+2C_X = 2(2l1+C_X), which makes t1 real and the argument goes through; alternatively the theorem should be restricted to l1 ≤ l2. The stated generality is therefore unsupported unless one of these repairs is made. The Fujita applications are not directly affected because the minimizers in m(C_X,l) have l1≤l2, but the theorem itself is the technical core and its stated generality is not supported.","section":"§4, Theorem 4.7"}],"minor_comments":[{"comment":"The parameter ε appears in the hypothesis without being quantified; the statement should say 'for every ε>0' (or 'for some ε>0') so that the strict inequality H² > 4(l1+l2+C_X) is well-defined.","section":"§4, Theorem 4.7"},{"comment":"In the definition of m′, the condition 'CX = 1' should read 'C_X = 1' for notational consistency.","section":"§1, Definition 1.4"},{"comment":"The text contains numerous typographical artifacts (e.g., 'surf-aces', 'K¨ a hler', 'M a sek') that should be cleaned before the final version.","section":"Global text"},{"comment":"The displayed degree expression in the proof, d_{s,t}(L⊗I_Y) = ... + l_Z ≥ 0, already uses the corrected t² formula; this inconsistency with the statement's t² should be resolved as part of the correction in Major Comment 1.","section":"§3, Proposition 3.9 proof"}],"recommendation":"major_revision","confidential_remarks":"The t² formula issue appears to be a repeated typo rather than a conceptual error, since the proof of Proposition 3.9 and the intended constants in the main theorems align with the corrected formula. The authors should be asked to fix it consistently and to check that Theorem 4.7 then covers all partitions. The external inputs from Langer and Koseki are used appropriately; there is no circularity. The paper is a good fit for the journal, and the main ideas are sound, but the written proof of the central technical theorem needs correction."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Anne's and Anda's paper is a genuine advance: it extends the Arcara–Bertram Bridgeland-stability approach to Reider-type vanishing on normal surfaces, covers positive characteristic via Langer's and Koseki's Bogomolov-type inequality, and treats the Du Bois canonical sheaf. The Fujita bounds for rational double points (global generation at a≥3, very ampleness at a≥4) are stated correctly and the proof strategy for those is coherent. The adaptation to non-line-bundle ω_X⊗L is new and worth having.\n\nThe paper has two real soft spots. First, Proposition 3.5 contains a numeric slip: with the stated t², the degree computation gives A·H ≤ 2ch₂ + 4rl + C_X r, not the printed 2ch₂ + 2rl. This looks like a simple sign/constant error, but it propagates into the Reider constants and should be fixed. Second, and more seriously, the proof of Theorem 4.7 does not cover partitions with l₁ > l₂. The argument at the stability parameter t₁ requires H² > 4(2l₁+C_X), while the theorem's hypothesis only gives H² ≥ max((C_X+2l₂+1)², 4(l₁+l₂+C_X)+ε). When l₁>l₂, the second term can be far smaller than 4(2l₁+C_X); for instance l₁=2, l₂=0, C_X=0 requires H² > 16 but the bound is only 8+ε. So the vanishing is unproved for every unbalanced partition as written. The Fujita applications use l₁≤l₂, so those corollaries survive, but Theorem 4.7 is the technical core and its stated generality is not supported. The fix is likely easy—restrict to l₁≤l₂, strengthen the H² bound to 4(2l₁+C_X)+ε when l₁>l₂, or find a different argument that avoids t₁.\n\nOne caveat to keep in mind: the advertised optimal constants depend on Langer's constant C_X, imported from [13] and Koseki's paper, not proved here. That is normal citation practice, but it means the numerical thresholds are conditional on that external inequality. I don't see circularity—the vanishing is derived from stability and the Bogomolov inequality, not assumed.\n\nThis paper deserves a serious referee. It has a real new result, and the gap is repairable. I would send it back for revision and check the repaired Theorem 4.7 carefully. Bring it to reading group if your group works on surfaces or stability.","headline":"Solid new Reider-type results on normal surfaces, but the main technical theorem has a repairable gap for unbalanced partitions that should be fixed before publication.","tokens_in":21621,"tokens_out":2480,"would_cite":true,"duration_ms":21510,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14J17","14F08","14C20","14J60"],"pacs":[],"model":"deepseek-v4-flash","headline":"Bridgeland stability on normal surfaces proves Reider-type jet separation for adjoint sheaves, recovering sharp bounds 3 and 4 for rational double point singularities.","keywords":["Reider-type theorem","Bridgeland stability","normal surfaces","jet separation","positive characteristic","Du Bois complex","Bogomolov inequality","adjoint linear series"],"falsifier":"On a concrete normal surface, compute the smallest admissible $C_X$ (for instance, on a cone over a plane curve, where the paper computes $C_X = 19/4$) and check whether there is an ample line bundle $L$ satisfying the theorem's inequalities, $L^2 \\geq \\max((C_X+2l_2+1)^2, 4(l_1+l_2+C_X)+\\varepsilon)$ and $L\\cdot C \\geq \\max(2(2l_1+C_X), C_X+2l_2+1)$ for all effective curves $C$, with $H^1(X, \\omega_X \\otimes L \\otimes I_Z) \\neq 0$ for some zero-dimensional subscheme $Z$; any such pair disproves the vanishing claim. Equivalently, a single slope-semistable sheaf violating the stated Bogomolov inequality for the claimed $C_X$ would break the stability argument at its base.","tokens_in":20474,"feed_emoji":"📐","tokens_out":13413,"duration_ms":115089,"temperature":0.7,"pith_summary":"This paper establishes Reider-type vanishing theorems on projective normal surfaces, including singular ones and surfaces over positive-characteristic fields, using Bridgeland stability conditions. The main numerical criterion says that if an ample line bundle $L$ on a normal surface $X$ satisfies certain quadratic and curve-intersection inequalities involving a surface-dependent constant $C_X$, then $H^1(X, \\omega_X \\otimes L \\otimes I_Z) = 0$ for every zero-dimensional subscheme $Z$; equivalently, sections of $\\omega_X \\otimes L$ separate jets along $Z$. From this the authors recover the optimal surface bounds predicted by the classical adjoint linear-series conjecture: $\\omega_X \\otimes L^a$ is globally generated for $a \\geq 3$ and very ample for $a \\geq 4$ whenever the constant $C_X$ vanishes, which happens for complex surfaces with at most rational double point singularities. The same mechanism works when $\\omega_X$ is replaced by the dual of the zeroth Du Bois complex, giving a Reider-type statement for a variant of the canonical sheaf adapted to singular surfaces.","feed_headline":"Vanishing theorem on singular surfaces proves sharp bounds 3 and 4","feed_subtitle":"A Bridgeland-stability argument extends Reider's vanishing to normal surfaces and positive characteristic.","key_machinery":"The engine is a recent construction of Bridgeland stability conditions on normal surfaces: each stability condition assigns a slope to complexes of sheaves through a central charge, and objects live in an abelian heart with Harder-Narasimhan filtrations. The central charge is modified by the constant $C_X$, which makes a Bogomolov-type inequality hold for slope-semistable sheaves. The paper isolates a class of objects 'of type O' — complexes whose cohomology is $O_X$ in degree $-1$, a zero-dimensional sheaf in degree $0$, and with no nonzero maps from any skyscraper sheaf — which contains both $O_X[1]$ and $\\Omega^0_X[1]$. For the stability condition with $s=1/2$ and a suitable $t$, objects of type $O$ and twisted ideal sheaves $L \\otimes I_Z$ are shown to be stable, so a nonzero morphism between them is impossible by Schur's lemma. The refinement is that the image of such a morphism, under weaker hypotheses, is a torsion sheaf whose support is an effective divisor $D$ satisfying Reider-type inequalities $D^2 < 1$ and $0 < D \\cdot L \\leq D^2 + C_X + l'$. Two length-redistribution lemmas move zero-dimensional length between $Z$ and $H^0(F)$ while preserving the total $l_Z + l_T$, which is what produces the optimal constants.","core_discovery":"The central discovery is that the smooth-surface strategy of proving Reider's theorem through Bridgeland stability extends to normal surfaces once the central charge is modified by a term controlled by a Bogomolov-type constant $C_X$. Concretely, Theorem 4.7 states: for any object $F$ of type $O$ (cohomology $O_X$ in degree $-1$, a zero-dimensional sheaf in degree $0$, and no nontrivial maps from skyscraper sheaves), and any decomposition $l_1 + l_2 = l_Z + \\mathrm{length}(H^0(F))$, the space $\\operatorname{Hom}(L \\otimes I_Z, F)$ vanishes whenever $L^2 \\geq \\max((C_X+2l_2+1)^2, 4(l_1+l_2+C_X)+\\varepsilon)$ and $L \\cdot C \\geq \\max(2(2l_1+C_X), C_X+2l_2+1)$ for every effective curve $C$. Taking $F = O_X[1]$ turns this into the vanishing $H^1(X, \\omega_X \\otimes L \\otimes I_Z)=0$ and hence into jet separation; taking $F = \\Omega^0_X[1]$ yields the Du Bois variant. The constants are sharp in the sense that for $C_X=0$, the choices $l_1=0, l_2=1$ and $l_1=1, l_2=1$ give respectively the bounds $a \\geq 3$ for global generation and $a \\geq 4$ for very ampleness.","pith_inferences":["The distribution of zero-dimensional length between $Z$ and $H^0(F)$ affects the bounds only through the total $l_Z+l_T$, suggesting a general principle: for any target complex whose cohomology is a sheaf plus a zero-dimensional piece, jet separation should depend only on the sum of the lengths, not on where the length sits.","The explicit positive-characteristic formulas for $C_X$ turn a qualitative failure of the fixed 3 and 4 bounds into a quantitative prediction: on a given surface, the smallest $a$ for which $\\omega_X \\otimes L^a$ separates jets can be computed as $m'(C_X, l_Z)$ once $C_X$ is known from the birational class.","The Du Bois variant suggests that the obstruction to good adjoint behavior on a singular complex surface has two independent sources, $C_X$ and the length of $H^1(\\Omega^0_X)$, so a combined 'singularity cost' $l_Z + l_T$ may be the right invariant in higher-dimensional analogues."],"forward_implications":["On any normal surface with $C_X=0$, in particular any complex surface with at most rational double point singularities, $\\omega_X \\otimes L^a$ separates jets for $a \\geq 3$ and is very ample for $a \\geq 4$, matching the sharp smooth-surface bounds.","In positive characteristic the theorem holds with degree bound $a \\geq m'(C_X, l_Z)$, where $C_X$ is explicitly known from the birational class; this explains why fixed bounds 3 and 4 cannot hold for all smooth surfaces in that setting.","The jet-separation statement does not require $K_X + L$ to be Cartier, so it applies when $\\omega_X \\otimes L$ is a sheaf rather than a line bundle.","For complex surfaces, replacing $\\omega_X$ by $\\omega_X^{\\mathrm{DB}}$, the derived dual of the zeroth Du Bois complex, yields the same Reider-type vanishing, with the length of $H^1(\\Omega^0_X)$ added to the subscheme length in all bounds.","Under the theorem's hypotheses, a nonzero morphism $L \\otimes I_Z \\to F$ forces an effective divisor $D$ with $D^2<1$ and $0<D\\cdot L \\leq D^2+C_X+l'$, which recovers the classical Reider criterion for very ampleness on smooth surfaces when $L^2>9$."],"supporting_citations":[{"why":"Supplies the Bridgeland stability conditions on normal surfaces and the Bogomolov-type inequality with constant $C_X$ that the theorem's numerical bounds depend on.","marker":"[13]"},{"why":"Provides the Chern character and intersection theory on normal varieties used to define the central charge and the Riemann-Roch identities.","marker":"[14]"},{"why":"Supplies the Bogomolov-Gieseker inequality in positive characteristic, giving the explicit $C_X$ values used in the positive-characteristic corollary.","marker":"[10]"},{"why":"The smooth-surface proof of Reider's theorem via Bridgeland stability that this paper generalizes to normal surfaces.","marker":"[1]"},{"why":"Used to show that $\\omega_X^{\\mathrm{DB}}$ is a subsheaf of $\\omega_X$, which is needed for the Du Bois complex variant.","marker":"[11]"},{"why":"Identifies $H^0(\\Omega^0_X)$ with the structure sheaf of the seminormalization, placing $\\Omega^0_X[1]$ in the type-O class used by the main theorem.","marker":"[18]"}],"fun_headline_variants":["Bridgeland stability proves Reider theorems on normal surfaces","Vanishing theorem on singular surfaces yields sharp bounds 3 and 4","Reider-type results for normal surfaces via Bridgeland stability","Positive characteristic Reider theorems on normal surfaces","Fujita bounds on singular surfaces from Bridgeland stability"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof relies on an external numerical inequality: each surface is assumed to carry a single constant $C_X$ that bounds how negative a certain quadratic expression attached to any slope-semistable sheaf can be; the optimal degrees 3 and 4 follow only when that constant is 0.","fun_headline_variants_meta":{"raw":{"variants":["Bridgeland stability proves Reider theorems on normal surfaces","Vanishing theorem on singular surfaces yields sharp bounds 3 and 4","Reider-type results for normal surfaces via Bridgeland stability","Positive characteristic Reider theorems on normal surfaces","Fujita bounds on singular surfaces from Bridgeland stability"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000274,"raw_usage":{"total_tokens":1650,"prompt_tokens":970,"completion_tokens":680,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":586,"completion_tokens_details":{"reasoning_tokens":596}},"tokens_in":586,"tokens_out":680,"duration_ms":20422,"temperature":1.0,"reasoning_tokens":596,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T21:04:48.612259+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"On a concrete normal surface, compute the smallest admissible $C_X$ (for instance, on a cone over a plane curve, where the paper computes $C_X = 19/4$) and check whether there is an ample line bundle $L$ satisfying the theorem's inequalities, $L^2 \\geq \\max((C_X+2l_2+1)^2, 4(l_1+l_2+C_X)+\\varepsilon)$ and $L\\cdot C \\geq \\max(2(2l_1+C_X), C_X+2l_2+1)$ for all effective curves $C$, with $H^1(X, \\omega_X \\otimes L \\otimes I_Z) \\neq 0$ for some zero-dimensional subscheme $Z$; any such pair disproves the vanishing claim. Equivalently, a single slope-semistable sheaf violating the stated Bogomolov inequality for the claimed $C_X$ would break the stability argument at its base.","supporting_citations":[{"cited_title":"Bridgeland stability conditions on normal surfa ces","cited_arxiv_id":null,"evidence_quote":"Supplies the Bridgeland stability conditions on normal surfaces and the Bogomolov-type inequality with constant $C_X$ that the theorem's numerical bounds depend on."},{"cited_title":"On the Bogomolov–Gieseker inequality in positive c haracteris- tic","cited_arxiv_id":null,"evidence_quote":"Supplies the Bogomolov-Gieseker inequality in positive characteristic, giving the explicit $C_X$ values used in the positive-characteristic corollary."},{"cited_title":"Reider’s theorem and Thadd eus pairs revisited","cited_arxiv_id":null,"evidence_quote":"The smooth-surface proof of Reider's theorem via Bridgeland stability that this paper generalizes to normal surfaces."},{"cited_title":"Du Bois singularities deform","cited_arxiv_id":null,"evidence_quote":"Used to show that $\\omega_X^{\\mathrm{DB}}$ is a subsheaf of $\\omega_X$, which is needed for the Du Bois complex variant."},{"cited_title":"Mixed Hodge complexes on algebraic varieties","cited_arxiv_id":null,"evidence_quote":"Identifies $H^0(\\Omega^0_X)$ with the structure sheaf of the seminormalization, placing $\\Omega^0_X[1]$ in the type-O class used by the main theorem."}],"review_version":1}