Derived category of coherent systems on curves and stability conditions

Aliaksandra Novik; Soheyla Feyzbakhsh

REVIEW 3 major objections 4 minor 46 references

Derived category of coherent systems on curves and stability conditions

T0 review · 3 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read This paper proves that the stability manifold of the derived category of coherent systems on a curve encodes the curve's Brill–Noether theory, by classifying every stability condition in an open locus as either a gluing of simpler stability

desk verdict Promising and structurally coherent, but the claimed equality 'stability manifold detects Brill–Noether' is not yet proved: the rational-to-real bridge in Theorem 3.3 is asserted, and even on the rational domain the proof needs w > Φ_C(b), not merely w > 0. read the letter →

arxiv 2511.01601 v2 pith:UFCAW4LS submitted 2025-11-03 math.AG

Soheyla Feyzbakhsh , Aliaksandra Novik This is my paper · ORCID

classification math.AG MSC 14H6014F08

keywords coherentsystemsBridgelandstabilityconditionsderivedcategoriesBrill–Noethertheorytiltinggluingwall-crossingcurves

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

The paper works with the bounded derived category of coherent systems on a smooth projective curve of genus g>0, where a coherent system is a triple consisting of a vector space, a sheaf, and a map between them. Unlike the derived category of coherent sheaves, whose stability manifold is a single point, this category admits an open locus of stability conditions with real geometric content. The paper proves a dichotomy: up to the GL+(2,R)-action, every stability condition in this open locus is either a gluing along a semiorthogonal decomposition (Type A) or a tilting stability condition sigma_{b,w} with w>Phi_C(b), where Phi_C is the Brill–Noether function of C (Type B). Consequently the quotient of one open component by GL+(2,R) is exactly the region in C above Phi_C, so the stability manifold detects the curve's Brill–Noether theory. This opens a route to Brill–Noether questions for vector bundles by wall-crossing inside a single fixed curve's derived category.

What carries the argument

The carrying object is the bounded derived category D(T_C) of coherent systems [V⊗O_C -> E], together with its two exceptional objects [O_C->0] and [O_C->O_C], which split the category via semiorthogonal decompositions into D(V) and D(C). The proof's main tool is the tilting construction: tilting the abelian category T_C at slope b with respect to the torsion pair (T_b,F_b) produces the heart A(b), and the Brill–Noether function Phi_C(x) governs when the central charge Z_{b,w} satisfies the positivity and support properties. For gluing, the standard gluing construction for stability conditions along a semiorthogonal decomposition yields stability conditions, with the inequality f(0)<1/2 enco

What would settle it

Exhibit a stability condition sigma in Stab°(D(T_C)) for which, after any GL+(2,R) action, two point objects [0->O_x] and [0->O_y] have different phases, or some [0->O_x] is strictly semistable; the paper provides no proof that this is impossible, and the case division in Lemmas 5.2–5.7 would not apply. Alternatively, for a specific curve (e.g. an elliptic curve), compute the quotient of the open locus by GL+(2,R) and check whether it equals the region above Phi_C; any discrepancy falsifies the classification.

Watch

Extended reading notes

Core claim

The central claim is Theorem 5.1: after a GL+(2,R) action, any stability condition sigma in Stab°(D(T_C))—meaning [O_C->0] and all [0->O_x] are sigma-stable—is either a gluing stability condition gl^{(1)}(sigma_V, sigma_g) with f(0)<1/2, or a tilting stability condition sigma_{b,w}=(A(b),Z_{b,w}) with w>Phi_C(b). Here A(b)=<F_b[1],T_b> is the heart obtained by tilting the category of coherent systems at slope b, Z_{b,w}(T)=-n(T)+w r(T)+ i(d(T)-br(T)), and Phi_C is the smallest upper-semicontinuous function bounding h^0(F)/rk(F) for semistable sheaves F. The corollary is that Stab°(D(T_C)) = U_A ∪ U_B, where U_B/GL+(2,R) = {b+iw : w>Phi_C(b)}; thus the stability manifold of the derived catego

Load-bearing premise

The entire classification rests on the unproved normalization—imported from a cited proposition in another paper—that after a GL+(2,R) rotation all objects [0->O_x] are stable of the same phase; if this fails for some stability condition in the open locus, the Type A/Type B dichotomy does not follow.

Editorial extensions

If this is right

If Theorem 5.1 holds, the stability manifold of D(T_C) is not a discrete point: the open locus contains a two-dimensional family U_B whose quotient identifies with the region above the Brill–Noether function of C.
The stability manifold therefore separates curves that share the same genus but have different Brill–Noether functions, a phenomenon impossible for the derived category of sheaves alone.
Within the two-dimensional slice (b,w), wall-and-chamber decomposition is locally finite, walls terminate on the boundary curve w=Phi_C(b), and the large-volume limit recovers classical alpha-stability of coherent systems.
A Bogomolov-type inequality holds for sigma_{b,w}-semistable objects and for alpha-semistable coherent systems, with an explicit convex domain U_f provided for curves of first Clifford index at least 2.
In the adjacent locus where [0->O_C] and the point sheaves are stable, every stability condition is a gluing of one of the two types, and the stable objects [0->E] are exactly shifts of slope-stable sheaves E.

Reading between the lines

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

One can read the quotient U_B/GL+(2,R) = {b+iw: w>Phi_C(b)} as a definition of the Brill–Noether function from stability data; this suggests that wall-crossing invariants in this slice could produce new proofs of non-emptiness or dimensional bounds for higher-rank Brill–Noether loci.
The dichotomy is likely only the first of several chambers: the second gluing family with f(0)>=1/2 suggests a larger atlas of the full stability manifold, possibly obtained by crossing the wall f(0)=1/2 and gluing along the other exceptional object.
Because Phi_C depends on the curve's Clifford index, the stability manifold provides a categorical invariant sensitive to special curves; one testable consequence is that curves with differing Clifford index cannot have isomorphic stability manifolds even if their derived categories of sheaves are equivalent.
A direct computational check on genus 1 or genus 2 curves, where Phi_C can be computed explicitly, would verify the quotient description and the chamber structure without relying on the normalization lemma.

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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

You should read this paper if you care about stability conditions on categories attached to curves. The main claim — that an open locus of Stab(D(T_C)) is a union of a gluing locus and a tilting locus whose wall boundary is the Brill–Noether function — is the right kind of statement, and the paper does real work: the setup of D(T_C), the Serre functor computation, the two semi-orthogonal decompositions, and the Type A/B classification in Section 5 are coherent and mostly argued with care. The rational-b tilt stability conditions (Lemmas 3.5 and 3.7) are the core technical contribution, and they look like genuine mathematics.

The soft spots are in the bridge from rational to real b. The text says Theorem 3.3 is proved only on the restricted domain (b,w) ∈ Q × R_{>0} and then asserts the full statement follows from Theorem 5.1 plus deformation theory. That is not a proof. Theorem 5.1 classifies stability conditions that are already assumed to exist; it does not construct one for every (b,w) in the region w > Φ_C(b). Bridgeland's deformation theorem gives a local homeomorphism around existing points, but one still has to show the set of realized parameters is open and closed in U_C, or otherwise bound the path. Without that, Corollary 5.8 gives only the inclusion from realized Type B conditions into {w > Φ_C(b)}, not the equality. The stress-test note is right about this. The main headline of the paper currently overstates what is established. The gap is not obviously fatal, but it is load-bearing.

Two smaller things. First, even on the rational domain, Lemma 3.5's proof needs w > Φ_C(μ(H^{-1}(T))), so the stated restricted domain w > 0 is too weak when b is positive; the proof as written works when w > Φ_C(b). This is a mismatch between statement and argument and should be fixed. Second, the normalization that all j_*O_x have the same phase is borrowed from FLZ22 by analogy. It is plausible and probably standard, but since the whole case division in Theorem 5.1 rests on it, a referee should ask for a proof or a precise citation. Proposition 6.1 is also stated without proof; if the wall-chamber description matters, that needs to be supplied or explicitly cited.

I would send this to a serious referee. The construction is valuable, the classification is plausible, and the gap is well-defined and may be closable. But the paper should not be accepted until the real-b extension is supplied or the theorem is honestly narrowed to rational b.

Referee Report

3 major / 4 minor

Summary. The paper studies the bounded derived category D(T_C) of (generalized) coherent systems on a smooth projective curve C and constructs two families of Bridgeland stability conditions: gluing stability conditions arising from the semiorthogonal decomposition into D(V) and D(C), and tilting stability conditions σ_{b,w} = (A(b), Z_{b,w}) with central charge Z_{b,w}(T) = -n(T) + w r(T) + i(d(T)-b r(T)). Defining a Brill-Noether function Φ_C(b) analogous to the Le Potier function, the authors claim that, up to the action of \widetilde{GL}^+(2,\mathbb{R}), every stability condition in the open locus Stab^\circ(D(T_C)) is either a gluing of Type A or a tilting stability condition of Type B with w > Φ_C(b). They further claim that the quotient U_B/\widetilde{GL}^+(2,\mathbb{R}) is exactly {b+iw : w > Φ_C(b)}, so the stability manifold detects the Brill-Noether function of C. The paper also studies the wall-and-chamber decomposition of the Type B slice, the large-volume limit recovering classical α-stability, and a second type of gluing stability condition.

Significance. If the main claims are correct, this is an interesting and nontrivial contribution: it gives an open subset of the stability manifold of a curve-derived category whose geometry is genuinely curve-dependent, through the Brill-Noether function, in contrast to the case of D^b(C) itself. The explicit description of Stab^\circ(D(T_C)), the identification of Type A and Type B loci, the wall/chamber analysis, and the connection to coherent-system stability are all valuable. The paper also benefits from a concrete algebraic description of D(T_C), including Serre and duality functors, and from a plausible overall architecture of the classification. However, several load-bearing steps remain insufficiently justified, in particular the passage from rational to arbitrary real b in the tilting family and the normalization of the skyscraper objects at the start of the classification. These gaps affect the central claim that the stability manifold encodes the full Brill-Noether function.

major comments (3)

[§3, Theorem 3.3] The theorem is stated for all (b,w) with w > Φ_C(b), but the proof is only given on the restricted domain (b,w) ∈ Q × R_{>0}. Even on that restricted domain the proof of Lemmas 3.5 and 3.7 actually requires w > Φ_C(b), not merely w > 0: Lemma 3.5 uses the inequality Φ_C(µ(H^{-1}(T))) < w, and Lemma 3.7 chooses δ with δ^{-1}(x-b)^2 + w_0 - δ > Φ_C(x), which is possible only if w_0 > Φ_C(b_0). Thus the restricted statement as written is not justified for 0 < w ≤ Φ_C(b). The closing sentence of §3 says the full real case follows from Theorem 5.1 together with deformation theory, but Theorem 5.1 classifies stability conditions that are assumed to exist; it does not construct a stability condition at every (b,w) with w > Φ_C(b). Bridgeland's deformation theorem only gives a local homeomorphism around existing stability conditions; to move from rational to irrational b one must show that the i
[§5, proof of Theorem 5.1, first paragraph] The classification begins with the assertion that, up to a \widetilde{GL}^+(2,\mathbb{R})-action, all point-sheaf objects j_*O_x can be made σ-stable of the same phase, justified only by 'a similar argument as in [FLZ22, Proposition 2.9]'. This normalization is load-bearing: the cohomological bounds in Lemmas 5.2–5.7, the torsion-pair analysis, the n=0/n≥1 dichotomy, and ultimately the Type A/Type B classification all depend on it. If this reduction fails for some σ in Stab^\circ(D(T_C)), the dichotomy is not established. The authors should either state and prove the analogue of [FLZ22, Proposition 2.9] in their setting or explain precisely how the cited result applies, since the category and the stability conditions here are not the same as in [FLZ22].
[§5, Case (I), proof of w > Φ_C(b)] In the paragraph after Lemma 5.5, the proof that w > Φ_C(b) says that since {w > Φ_C(b)} is open, by deformation theory 'it suffices to prove the claim for b ∈ Q'. This reduction is not valid as stated: deformation theory is a local statement around existing stability conditions, and the proof for rational b does not show that every irrational b occurring in the Type B locus can be connected to a rational b by a path lying in the image of Stab. This is the same rational-to-real gap as in Major Comment 1, and it directly affects Corollary 5.8: without a proof that σ_{b,w} is defined for all (b,w) with w > Φ_C(b), the asserted equality U_B/\widetilde{GL}^+(2,\mathbb{R}) = {b+iw : w > Φ_C(b)} is only an inclusion from realized Type B conditions into that region, not the claimed bijection.

minor comments (4)

[§7, proof of Theorem 7.1] The sentence 'such that j_*O_x and j_*O_x are σ-stable for all points x ∈ C' should read '[0→O_C] and [0→O_x] are σ-stable'. The same typo appears in the statement of Proposition 7.2's context.
[Throughout] The notation \widetilde{GL}^+(2,\mathbb{R}) appears in the text as 'fGL^+(2,R)' in several places. Please use a consistent mathematical rendering.
[§6, Proposition 6.1] The wall-and-chamber structure is stated with a proof that is omitted entirely ('the argument is identical ... we omit the repetition'). Since the subsequent Propositions 6.2, 6.5, and 6.6 rely on this structure, a fuller proof or a precise reference with the necessary adaptations would improve the paper.
[§3, Lemma 3.2] The proof that the Brill-Noether function Φ_C is well-defined, upper semicontinuous, and satisfies the stated bounds is very brief. A reference for the limiting construction and for the upper-semicontinuity property would be helpful.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the Brill–Noether function is an external invariant, and the matching bound is derived from stability itself; the real-b extension is a rigor gap, not a circular reduction.

full rationale

I find no circular step in the claimed derivation. The Brill–Noether function Φ_C is defined directly from the curve via h^0/rk on semistable sheaves, before any stability condition is constructed (Section 3, after Definition 3.1); it is not fitted to Stab(D(T_C)). In Theorem 5.1, Case (I), the bound w > Φ_C(b) is derived from stability: for a slope-stable sheaf E of slope b, the object [O_C⊗H^0(E)→E][1] lies in the heart and its real central charge equals -rk(E)w + h^0(E) < 0, giving h^0(E)/rk(E) < w. This is an independent necessity argument, not an input of the construction. The sufficiency direction on rational b is proved in Lemmas 3.5 and 3.7; the threshold appears because positivity of the central charge forces w to dominate the defining Brill–Noether bound, which is the same external invariant rather than a renamed prediction. The only flagged issue is the extension to all real b. The paper itself states: 'In this section, we prove the claim only on the restricted domain (b, w)∈Q×R_{>0}; Lemma 3.5 proves they are pre-stability conditions, and Lemma 3.7 verifies the support property. The theorem then follows from the classification in Theorem 5.1 together with the deformation theory of Bridgeland stability conditions [Bri07, Theorem 1.2] or [Bay16, Theorem 1.2].' This is an acknowledged omitted argument: a classification theorem about existing stability conditions does not by itself construct new ones at irrational b, and deformation theory alone does not establish the required openness/closedness along paths to the real boundary. That affects the completeness of the claimed equality U_B/GL^+(2,R) = {w > Φ_C(b)} in Corollary 5.8, but it is a correctness/rigor gap, not a case in which an output reduces to an input by definition or a fitted parameter is relabelled as a prediction. No load-bearing self-citation occurs: the normalization that j_*O_x are stable of the same phase is cited to the non-author work [FLZ22, Proposition 2.9]; the authors' own [FT21] is cited only for a standard wall-crossing argument, and [FL21] is contextual. The central classification and the external curve invariant give the result independent content.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No curve-specific constants are fitted; the 'Brill-Noether function' is defined from h^0 statistics of semistable bundles and is an input to the construction, not an output of a fit. The ledger is dominated by background theorems and one unproved normalization step.

assumptions (6)

domain assumption The category T_C is noetherian and artinian, so [Rud97, Theorem 2] yields Harder-Narasimhan filtrations for μ-stability.
Used to define the torsion pair (T_b,F_b) and heart A(b) in Section 3; no proof is given in the paper.
standard math Macrì's classification Stab(D(C)) ≅ GL+(2,R) [Mac07].
Used in the gluing constructions in Section 4 and in the quotient description in Corollary 5.8.
standard math Collins-Polishchuk gluing theorem [CP10, Theorem 3.6] and converse [CP10, Proposition 2.2].
Used for both gluing types and to identify glued stability conditions in Sections 4 and 5.
ad hoc to paper All skyscraper objects j_*O_x can be made σ-stable of the same phase via a GL+(2,R) action, by analogy with [FLZ22, Proposition 2.9].
Unproved in this paper; load-bearing for Lemma 5.2 and the trichotomy in Theorem 5.1.
standard math Existence of slope-stable bundles of every rational slope, and Clifford's theorem bound for h^0(E)/rank(E).
Used in Lemma 3.2 to show Φ_C is well-defined and to prove w>Φ_C(b).
domain assumption Equivalence Ku(blow-up) ≃ D(T_C) from [AK25].
Used in Section 2.3 for the dual functor and for motivation; not visibly needed for the classification proof.

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Pith. "Pith review of Derived category of coherent systems on curves and stability conditions." pith.science (2026). https://pith.science/paper/UFCAW4LS

@misc{pith2026251101601,
  author       = {Pith},
  title        = {Pith review of: Derived category of coherent systems on curves and stability conditions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UFCAW4LS}},
  note         = {Machine review of arXiv:2511.01601}
}

abstract

Let $C$ be a smooth projective curve of genus $g>0$. We describe an open locus of Bridgeland stability conditions on the bounded derived category of coherent systems on $C$, and show that stability manifold detects the Brill--Noether theory of the curve.

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Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.

[1] [1]

Alexeev and A

V. Alexeev and A. Kuznetsov. Augmentations, reduced ideal point gluings and compact type degenerations of curves. arXiv e-Prints , 2025, 2509.12429

arXiv 2025

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E. Ballico. Coherent systems of type (r,d,r+1) on general nodal or cuspidal curves. Int. J. Pure Appl. Math. , 48(4):517--522, 2008

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