Reider-type theorems on normal surfaces via Bridgeland stability

Anda Tenie; Anne Larsen

REVIEW 2 major objections 4 minor 22 references

Reider-type theorems on normal surfaces via Bridgeland stability

T0 review · 2 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Bridgeland stability on normal surfaces proves Reider-type jet separation for adjoint sheaves, recovering sharp bounds 3 and 4 for rational double point singularities.

desk verdict 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. read the letter →

arxiv 2411.09107 v1 pith:2THRD2TD submitted 2024-11-14 math.AG

Anne Larsen , Anda Tenie This is my paper · ORCID

classification math.AG MSC 14J1714F0814C2014J60

keywords Reider-typetheoremBridgelandstabilitynormalsurfacesjetseparationpositivecharacteristicDuBoiscomplexBogomolovinequalityadjointlinearseries

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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 $L^2>9$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the 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.

The 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.

One 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.

This 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.

Referee Report

2 major / 4 minor

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).

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 (2)

[§3, Proposition 3.5 and Definition of t] 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).
[§4, Theorem 4.7] 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.

minor comments (4)

[§4, Theorem 4.7] 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.
[§1, Definition 1.4] In the definition of m′, the condition 'CX = 1' should read 'C_X = 1' for notational consistency.
[Global text] The text contains numerous typographical artifacts (e.g., 'surf-aces', 'K¨ a hler', 'M a sek') that should be cleaned before the final version.
[§3, Proposition 3.9 proof] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Reider-type vanishing theorems are derived from external Bogomolov inequalities and Bridgeland stability, with no fitted parameter renamed as a prediction.

full rationale

The main derivation chain is self-contained relative to its stated external inputs. Theorem 4.7 proves Hom(L⊗I_Z,F)=0 by assuming length data l1,l2 with l1+l2=lZ+lT and imposing numerical bounds on H² and H·C. Those bounds are expressed through C_X, which is imported from Langer's Bogomolov-type inequality on normal surfaces [13] (and Koseki's positive-characteristic formulas [10]); C_X is not fitted to any Reider-type conclusion, and the paper does not use the target vanishing to define the stability conditions or the constants. The Bridgeland stability conditions are constructed as in Langer's work, and the stability arguments adapt Arcara–Bertram [1], both prior independent results rather than self-citations by the present authors. Propositions 4.2–4.3 only redistribute the finite lengths between Z and H⁰(F) while preserving lZ+lT; this is a structural reduction, not a concealed restatement of the desired vanishing. The optimal Fujita constants 3 and 4 are obtained by optimizing the explicit functions m(C_X,l) and m'(C_X,l), not by calibrating constants to force the answer. No equation in the paper is defined in terms of the target result, and no fitted input is later renamed as a prediction. The skeptical concern that Theorem 4.7 is not proved for partitions with l1>l2, because the stated H² hypothesis may be too weak to make the stability condition t1 real, is a proof-gap or correctness issue rather than a circularity issue: it does not show that the theorem's conclusion is equivalent to its hypotheses or to a self-citation chain. Under the circularity rubric, this does not raise the score. Accordingly, the paper merits a non-circularity finding of 0.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new physical entities; its nonstandard input is the surface-dependent constant C_X and the class of 'type O' objects, which is a definition rather than an invented entity. All major tools are cited from prior work.

free parameters (1)

C_X = surface-dependent, e.g., 0 for smooth/RDP char 0; explicit formulas in Cor 1.9; 19/4 in Ex 2.2
The Bogomolov constant that enters every threshold in the Reider-type bounds. It is not fitted to the vanishing conclusion, but the paper's numerical results depend on its value and it is imported from Langer/Koseki.

assumptions (4)

domain assumption Langer's construction of Bridgeland stability conditions on normal surfaces yields a heart and central charge with the stated rank and degree formulas.
Used throughout §3 as the framework for stability; cited from [13] rather than proven.
domain assumption Bogomolov-type inequality ∫(ch1(E)^2 - 2ch0(E)ch2(E) + C_X ch0(E)^2) ≥ 0 holds for ω-slope semistable torsion-free sheaves with a constant C_X as described.
Used in Props 3.5, 3.9, 3.10; the entire Reider threshold is expressed through C_X. Cited from [13] and [10].
domain assumption For a complex normal surface, Ω^0_X[1] is of type O, including Hom(Op,Ω^0_X[1])=0 and D(Ω^0_X) being a sheaf.
Used for Theorem 1.12 and Prop 4.1; relies on Du Bois complex theory from [2], [11], [18], [20] and Prop 2.4.
standard math The Hodge index theorem holds for the Mumford intersection product on A_1(X) (Observation 2.1(3)).
Used in Lemma 3.6 and Prop 3.9 to bound divisor self-intersections.

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Cite this review

Pith. "Pith review of Reider-type theorems on normal surfaces via Bridgeland stability." pith.science (2026). https://pith.science/paper/2THRD2TD

@misc{pith2026241109107,
  author       = {Pith},
  title        = {Pith review of: Reider-type theorems on normal surfaces via Bridgeland stability},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2THRD2TD}},
  note         = {Machine review of arXiv:2411.09107}
}

abstract

Using Langer's construction of Bridgeland stability conditions on normal surfaces, we prove Reider-type theorems generalizing the work done by Arcara-Bertram in the smooth case. Our results still hold in positive characteristic or when $\omega_X \otimes L$ is not necessarily a line bundle. They also hold when the dualizing sheaf is replaced by a variant arising from the theory of Du Bois complexes. For complex surfaces with at most rational double point singularities, we recover the optimal bounds for global generation and very ampleness as predicted by Fujita's conjecture.

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [13]

Bridgeland stability conditions on normal surfa ces

Adrian Langer. “Bridgeland stability conditions on normal surfa ces”. In: An- nali di Matematica Pura ed Applicata (1923-) (2024), pp. 1–12

work page 2024

[2] [1]

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Daniele Arcara and Aaron Bertram. “Reider’s theorem and Thadd eus pairs revisited”. In: Clay Math. Proc. 14 (2011), pp. 51–68

work page 2011

[3] [2]

Complexe de de Rham ﬁltr´ e d’une vari´ et´ e singuli` ere

Philippe Du Bois. “Complexe de de Rham ﬁltr´ e d’une vari´ et´ e singuli` ere”. In: Bulletin de la Soci´ et´ e math´ ematique de France109 (1981), pp. 41–81

work page 1981

[4] [3]

Global generation of plur icanonical and adjoint linear series on smooth projective threefolds

Lawrence Ein and Robert Lazarsfeld. “Global generation of plur icanonical and adjoint linear series on smooth projective threefolds”. In: Journal of the American Mathematical Society 6.4 (1993), pp. 875–903

work page 1993

[5] [4]

Global G eneration Of Linear Series On Terminal Threefolds

Lawrence Ein, Robert Lazarsfeld, and Vladimir Ma¸ sek. “Global G eneration Of Linear Series On Terminal Threefolds”. In: International Journal of Math- ematics 6.01 (1995), pp. 1–18

work page 1995

[6] [5]

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Takao Fujita. “Contribution to birational geometry of algebraic varieties: open problems”. In: the 23rd International Symposium, Division of Mathematics , the Taniguchi Foundation; August 22-27, 1988, Katata . 1988. REFERENCES 24

work page 1988

[7] [6]

Intersection theory

William Fulton. Intersection theory. Vol. 2. Springer Science & Business Me- dia, 2013

work page 2013

[8] [7]

Counterexamples to Fu jita’s con- jecture on surfaces in positive characteristic

Yi Gu, Lei Zhang, and Yongming Zhang. “Counterexamples to Fu jita’s con- jecture on surfaces in positive characteristic”. In: Advances in Mathematics 400 (2022), p. 108271

work page 2022