Moduli of stable sheaves on quadric threefold

Abstract

For each $0<\alpha<\frac{1}{2}$, there exists a Bayer--Lahoz--Macr{\`{\i}}--Stellari inducing Bridgeland stability condition $\sigma(\alpha)$ on a Kuznetsov component $\mathrm{Ku}(Q)$ 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 $\mathrm{Ku}(Q)$ with the numerical class $[\mathcal{P}_{x}]$, where $\mathcal{P}_{x}\in \mathrm{Ku}(Q)$ is the projection sheaf of the skyscraper sheaf at a closed point $x\in Q$. We show that the moduli space $\overline{M}_{Q}(\mathbf{v})$ of Gieseker semistable sheaves with Chern character $\mathbf{v}=\mathrm{ch}(\mathcal{P}_{x})$ is smooth and irreducible of dimension four, and prove that the moduli space $M_{\sigma(\alpha)}([\mathcal{P}_{x}])$ is isomorphic to $\overline{M}_{Q}(\mathbf{v})$. 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 $Q$.