Moduli of stable sheaves on quadric threefold

For each $0<\alpha <\frac{1}{2}$, there exists a Bayer–Lahoz–Macr{ì}–Stellari inducing Bridgeland stability condition $\sigma (\alpha )$ on a Kuznetsov component $\mathrm{K}\mathrm{u}(Q)$ of the smooth quadric threefold $Q$. We obtain the non-emptiness of the moduli space $M_{\sigma (\alpha )}([{\mathcal{P}}_x])$ of $\sigma (\alpha )$-semistable objects in $\mathrm{K}\mathrm{u}(Q)$ with the numerical class $[{\mathcal{P}}_x]$, where ${\mathcal{P}}_x\in \mathrm{K}\mathrm{u}(Q)$ is the projection sheaf of the skyscraper sheaf at a closed point $x\in Q$. We show that the moduli space ${\bar{M}}_Q(\mathbf{v})$ of Gieseker semistable sheaves with Chern character $\mathbf{v}=\mathrm{c}\mathrm{h}({\mathcal{P}}_x)$ is smooth and irreducible of dimension four, and prove that the moduli space $M_{\sigma (\alpha )}([{\mathcal{P}}_x])$ is isomorphic to ${\bar{M}}_Q(\mathbf{v})$. As an application, we show that the quadric threefold $Q$ can be reinterpreted as a Brill–Noether locus in the Bridgeland moduli space $M_{\sigma (\alpha )}([{\mathcal{P}}_x])$. In the appendices, we show that the moduli space $M_{\sigma (\alpha )}([S])$ contains only one single point corresponding to the spinor bundle $S$ and give a Bridgeland moduli interpretation for the Hilbert scheme of lines in $Q$.