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 $\sigma_{ \widetilde{ \mathcal{ D}}'}$ on its categorical resolution $\widetilde{ \mathcal{ D}}' \subset \mathrm{ D^b}(\widetilde{ X})$, which is compatible with the Verdier localization $\mathbf{ R}\pi_\ast$ 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 $\pi \colon \widetilde{ X} \longrightarrow X$ 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 $\sigma_{ \widetilde{ \mathcal{ D}}'}$.
Categories: math.AG