Stability conditions on a singular quadric threefold
Tzu-Yang Chou
TL;DR
The paper addresses stability conditions on the Kuznetsov component $\mathrm{Ku}(X)$ of a singular quadric threefold $X$ with a single ordinary double point by constructing a weak stability on a categorical resolution $\widetilde{\mathcal{D}}'$ and descent to $\mathrm{Ku}(X)$ via a Verdier localization. The method hinges on two semiorthogonal decompositions of the ambient $\mathrm{D^b}(\widetilde{X})$, the extraction of a full Ext-exceptional collection of length $3$ in $\widetilde{\mathcal{D}}'$, and a carefully tilted heart $\widetilde{\mathcal{A}}$ that descends to a heart $\mathcal{A}$ on $\mathrm{Ku}(X)$; together with a localization-compatible central charge, this yields a stability condition on $\mathrm{Ku}(X)$ and a compatible weak stability on $\widetilde{\mathcal{D}}'$. The construction relies on Kuznetsov’s categorical resolution, the projective-bundle geometry of the resolution $\widetilde{X}$, and the framework for descending hearts through the categorical absorption of singularities (KS24). This work extends prior two-dimensional results to a threefold setting and exemplifies a systematic route to stability conditions on singular Kuznetsov components.
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 $σ_{\widetilde{\mathcal{D}}'}$ on its categorical resolution $\widetilde{\mathcal{D}}' \subset \mathrm{D^b}(\widetilde{X})$, which is compatible with the Verdier localization $\mathbf{R}π_\ast$ and descends to a Bridgeland stability condition on $\mathcal{K}u(X)$. This can be viewed as a three-dimensional analogue of our previous result. We describe the geometry of the blow-up $π\colon \widetilde{X} \longrightarrow X$ 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 $\widetilde{\mathcal{D}}' \subseteq \mathrm{D^b}(\widetilde{X})$ resolving $\mathcal{K}u(X)$ and show that it admits a full Ext-exceptional collection, from which we construct $σ_{\widetilde{\mathcal{D}}'}$.