Stability conditions on a singular quadric threefold

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.