Stability conditions on a singular quadric threefold

Tzu-Yang Chou

arxiv: 2511.20164 · v2 · submitted 2025-11-25 · 🧮 math.AG

Stability conditions on a singular quadric threefold

Tzu-Yang Chou This is my paper

Pith reviewed 2026-05-17 05:05 UTC · model grok-4.3

classification 🧮 math.AG

keywords singular quadric threefoldKuznetsov componentcategorical resolutionBridgeland stability conditionsemiorthogonal decompositionderived categoryblow-upordinary double point

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The pith

Weak stability condition on categorical resolution of singular quadric threefold induces Bridgeland stability on derived category

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

The paper constructs a weak stability condition on Kuznetsov's categorical resolution inside the derived category of the blow-up of a singular quadric threefold. This condition is made compatible with the Verdier localization map that pushes forward to the derived category of the original singular variety. Compatibility ensures the weak condition descends to a full Bridgeland stability condition on D^b(X). The same construction restricts to give a stability condition on the Kuznetsov component and its own categorical resolution. A reader would care because these conditions organize objects in derived categories and support the study of moduli spaces for threefolds with mild singularities.

Core claim

We construct a weak stability condition on Kuznetsov's categorical resolutionwidetilde{D} subset D^b(widetilde{X}), compatible with the Verdier localization R pi_* : widetilde{D} to D^b(X), and hence obtain a Bridgeland stability condition on D^b(X). Restricting the construction, we obtain the corresponding statement for Ku(X) and its categorical resolution widetilde{D}'. We describe the geometry of the blow-up and obtain two semiorthogonal decompositions of D^b(widetilde{X}), arising from the projective bundle structure and from Kuznetsov's categorical resolution. Comparing them isolates an admissible subcategory that admits a full Ext-exceptional collection from which the localization-

What carries the argument

Comparison of the two semiorthogonal decompositions of D^b(widetilde{X}) (one from the projective bundle structure of the blow-up and one from Kuznetsov's categorical resolution) to isolate an admissible subcategory admitting a full Ext-exceptional collection

Load-bearing premise

The semiorthogonal decompositions from the projective bundle structure and Kuznetsov's categorical resolution can be compared to isolate an admissible subcategory that admits a full Ext-exceptional collection and supports a localization-compatible weak stability condition.

What would settle it

An explicit computation exhibiting an object whose stability phase cannot be consistently assigned under the descended condition, or proving that the isolated subcategory lacks a full Ext-exceptional collection, would disprove the descent.

read the original abstract

Let $X \subset \mathbb{P}^4$ be a quadric threefold with a single ordinary double point, and let $\mathcal{K}u(X)$ be its Kuznetsov component. In this paper, we construct a weak stability condition on Kuznetsov's categorical resolution $\widetilde{D} \subset \mathrm{D^b}(\widetilde{X})$, compatible with the Verdier localization $\mathbf{R}\pi_* \colon \widetilde{D} \to \mathrm{D^b}(X)$, and hence obtain a Bridgeland stability condition on $\mathrm{D^b}(X)$. Restricting the construction, we obtain the corresponding statement for $\mathcal{K}u(X)$ and its categorical resolution $\widetilde{D}'$. These can be viewed as a three-dimensional analogue of our previous result in \cite{Cho25}. We describe the geometry of the blow-up $\pi \colon \widetilde{X} \to X$ and obtain two semiorthogonal decompositions of $\mathrm{D^b}(\widetilde{X})$, arising from the projective bundle structure of $\widetilde{X}$ and from Kuznetsov's categorical resolution. Comparing them, we isolate an admissible subcategory $\widetilde{\mathcal{D}}\subset \mathrm{D^b}(\widetilde{X})$ resolving $\mathrm{D^b}(X)$ and show that it admits a full Ext-exceptional collection, from which we construct the localization-compatible weak stability condition.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Chou constructs a localization-compatible weak stability condition on the categorical resolution of the Kuznetsov component for a nodal quadric threefold by comparing two SODs and extracting an admissible subcategory with an Ext-exceptional collection.

read the letter

The punchline is that this paper delivers an explicit three-dimensional case of a weak stability condition on the resolved Kuznetsov component that descends under Verdier localization to give a Bridgeland stability condition on the derived category of the singular quadric itself. It is framed as the direct analogue of the author's earlier two-dimensional result, and the construction appears to go through for this specific geometry. What is actually new is the isolation of the admissible subcategory by aligning the projective-bundle semiorthogonal decomposition with the one coming from Kuznetsov's resolution, then verifying that the resulting piece carries a full Ext-exceptional collection from which the stability data can be built. The geometric input from the blow-up at the node supplies independent control that keeps the argument from collapsing into pure formal manipulation. The paper does this cleanly enough that a reader familiar with the standard SODs for quadrics and with the two-dimensional precedent can follow the steps without extra machinery. The soft spots are modest. The abstract and sketch leave the explicit verification of the support property and the phase separation for objects in the kernel of the localization functor somewhat implicit, so one has to check the full write-up to confirm that no hidden assumptions about the central charge or the heart are needed for the descent to work. The reliance on the author's prior paper for the two-dimensional case is fine, but it means the three-dimensional argument is not fully self-contained for someone who has not read the earlier work. Overall the central claim holds up on the evidence given. This paper is for people already working on Bridgeland stability conditions, semiorthogonal decompositions, and categorical resolutions of singular threefolds. A reader who knows the Kuznetsov component literature and the basic facts about nodal quadrics will extract the most value; it is not an entry point for outsiders. It deserves a serious referee because the construction is concrete, the claim is checkable, and the field gains from having explicit examples at this dimension even if some details need tightening in revision.

Referee Report

2 major / 2 minor

Summary. The paper constructs a weak stability condition on Kuznetsov's categorical resolutionwidetilde{D} subset D^b(widetilde{X}) for a quadric threefold X with a single ordinary double point. This condition is compatible with the Verdier localization R pi_* : widetilde{D} to D^b(X), inducing a Bridgeland stability condition on D^b(X). Restricting the construction yields the analogous result for the Kuznetsov component Ku(X) and its resolution widetilde{D}'. The approach proceeds by describing the blow-up pi : widetilde{X} to X, obtaining two semiorthogonal decompositions of D^b(widetilde{X}) (one from the projective bundle structure and one from Kuznetsov's resolution), comparing them to isolate an admissible subcategory widetilde{D} that admits a full Ext-exceptional collection, and constructing the localization-compatible weak stability condition from this collection. The result is framed as a three-dimensional analogue of the authors' prior work.

Significance. If the central construction holds, this provides an explicit example of a Bridgeland stability condition on the derived category of a singular threefold via categorical resolution, extending the authors' previous result on surfaces to dimension three. The use of standard results on quadric threefolds, semiorthogonal decompositions, and exceptional collections, combined with the explicit isolation of an admissible subcategory carrying a full Ext-exceptional collection, strengthens the geometric input and offers a template for similar constructions on other singular varieties.

major comments (2)

[Comparison of semiorthogonal decompositions] In the section comparing the two semiorthogonal decompositions of D^b(widetilde{X}), the manuscript must explicitly verify that the orthogonal complements align to produce an admissible subcategory widetilde{D} such that (i) widetilde{D} resolves D^b(X) via R pi_*, (ii) widetilde{D} carries a full Ext-exceptional collection, and (iii) the induced weak stability condition satisfies the support property with objects in ker(R pi_*) receiving phases outside the relevant interval. The current outline does not provide this verification.
[Construction of the weak stability condition] In the construction of the weak stability condition from the Ext-exceptional collection, the paper should include a direct check that the pair (heart, central charge) satisfies all axioms of a weak stability condition and descends under the Verdier localization without additional assumptions on the phases or the kernel of R pi_*.

minor comments (2)

[Notation and statements] Clarify the precise relationship between widetilde{D} and widetilde{D}' in the statements for D^b(X) and Ku(X) to avoid potential notational confusion.
[References] Expand the reference to the prior work cited as Cho25 with full bibliographic details.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We are grateful to the referee for the detailed and insightful report. The comments highlight areas where the manuscript can be strengthened by providing more explicit verifications. We address each major comment below and plan to incorporate the necessary additions in the revised version.

read point-by-point responses

Referee: [Comparison of semiorthogonal decompositions] In the section comparing the two semiorthogonal decompositions of D^b(widetilde{X}), the manuscript must explicitly verify that the orthogonal complements align to produce an admissible subcategory widetilde{D} such that (i) widetilde{D} resolves D^b(X) via R pi_*, (ii) widetilde{D} carries a full Ext-exceptional collection, and (iii) the induced weak stability condition satisfies the support property with objects in ker(R pi_*) receiving phases outside the relevant interval. The current outline does not provide this verification.

Authors: We thank the referee for this observation. We agree that the comparison section would benefit from more explicit verification of the alignment of the orthogonal complements. In the revised manuscript we will expand the relevant section with detailed arguments confirming that the complements produce an admissible subcategory widetilde{D} satisfying (i) resolution of D^b(X) via R pi_*, (ii) the existence of a full Ext-exceptional collection, and (iii) the support property of the induced weak stability condition with objects in ker(R pi_*) lying outside the relevant phase interval. revision: yes
Referee: [Construction of the weak stability condition] In the construction of the weak stability condition from the Ext-exceptional collection, the paper should include a direct check that the pair (heart, central charge) satisfies all axioms of a weak stability condition and descends under the Verdier localization without additional assumptions on the phases or the kernel of R pi_*.

Authors: We acknowledge the value of a direct verification. In the revision we will add an explicit step-by-step check that the pair consisting of the heart and central charge satisfies the axioms of a weak stability condition and that the construction descends under the Verdier localization R pi_*, using only the properties of the Ext-exceptional collection without imposing further assumptions on phases or the kernel. revision: yes

Circularity Check

1 steps flagged

Minor self-citation to prior analogue work; central steps rely on independent blow-up geometry and SOD comparison rather than reducing to inputs by construction.

specific steps

self citation load bearing [Abstract]
"These can be viewed as a three-dimensional analogue of our previous result in cite{Cho25}."

The paper frames its main construction as a direct three-dimensional analogue of the author's own prior work, so that the method for producing a localization-compatible weak stability condition from an Ext-exceptional collection is imported rather than re-derived from the new geometric input.

full rationale

The derivation begins with the explicit geometry of the blow-up π: X̃ → X at the node, produces two SODs (projective bundle and Kuznetsov resolution), and compares them to isolate an admissible subcategory D̃ with a full Ext-exceptional collection. From this collection a weak stability condition is built and shown compatible with Rπ_*. These steps use standard facts about quadrics and semiorthogonal decompositions plus the concrete blow-up data; they do not reduce the stability axioms or localization compatibility to a fitted parameter or self-referential definition. The single reference to Cho25 is presented only as an analogy and is not invoked to justify uniqueness or to close any gap in the present argument.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard background results in algebraic geometry and derived categories rather than introducing new free parameters or invented entities. The central claim depends on the existence and properties of the blow-up, the semiorthogonal decompositions, and the definition of weak stability conditions.

axioms (2)

domain assumption The blow-up of an ordinary double point on a quadric threefold yields a smooth projective bundle structure whose derived category admits a semiorthogonal decomposition.
Invoked in the description of the geometry of pi : widetilde{X} to X and the comparison of decompositions.
domain assumption Kuznetsov's categorical resolution exists and is compatible with Verdier localization from the resolved category to the derived category of the singular space.
Central to the construction of the weak stability condition on widetilde{D}.

pith-pipeline@v0.9.0 · 5531 in / 1588 out tokens · 31845 ms · 2026-05-17T05:05:12.410591+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

2 extracted references · 2 canonical work pages · 1 internal anchor

[1]

[Bri07] Tom Bridgeland

arXiv:2208.03173. [Bri07] Tom Bridgeland. Stability conditions on triangulated categories.Ann. of Math. (2), 166(2):317–345,

work page internal anchor Pith review arXiv
[2]

14 TZU-YANG CHOU [Orl92] D. O. Orlov. Projective bundles, monoidal transformations, and derived categories of coherent sheaves.Izv. Ross. Akad. Nauk Ser. Mat., 56(4):852–862, 1992

work page 1992

[1] [1]

[Bri07] Tom Bridgeland

arXiv:2208.03173. [Bri07] Tom Bridgeland. Stability conditions on triangulated categories.Ann. of Math. (2), 166(2):317–345,

work page internal anchor Pith review arXiv

[2] [2]

14 TZU-YANG CHOU [Orl92] D. O. Orlov. Projective bundles, monoidal transformations, and derived categories of coherent sheaves.Izv. Ross. Akad. Nauk Ser. Mat., 56(4):852–862, 1992

work page 1992