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 $\tilde{D}\subset {\mathrm{D}}^{\mathrm{b}}(\tilde{X})$, compatible with the Verdier localization $\mathbf{R}{\pi }_{\ast }\text{\hspace{0.17em}⁣}:\tilde{D}\to {\mathrm{D}}^{\mathrm{b}}(X)$, and hence obtain a Bridgeland stability condition on ${\mathrm{D}}^{\mathrm{b}}(X)$. Restricting the construction, we obtain the corresponding statement for $\mathcal{K}u(X)$ and its categorical resolution ${\tilde{D}}^{\mathrm{\prime }}$. These can be viewed as a three-dimensional analogue of our previous result in \cite{Cho25}. We describe the geometry of the blow-up $\pi \text{\hspace{0.17em}⁣}:\tilde{X}\to X$ and obtain two semiorthogonal decompositions of ${\mathrm{D}}^{\mathrm{b}}(\tilde{X})$, arising from the projective bundle structure of $\tilde{X}$ and from Kuznetsov’s categorical resolution. Comparing them, we isolate an admissible subcategory $\tilde{\mathcal{D}}\subset {\mathrm{D}}^{\mathrm{b}}(\tilde{X})$ resolving ${\mathrm{D}}^{\mathrm{b}}(X)$ and show that it admits a full Ext-exceptional collection, from which we construct the localization-compatible weak stability condition.