Moduli of stable sheaves on quadric threefold

For each 0<<12, there exists a Bayer--Lahoz--Macr\`{}--Stellari's inducing Bridgeland stability condition () on a Kuznetsov component Ku (Q) of the smooth quadric threefold Q. We obtain the non-empty of the moduli space M () (Pₗ) of () -semistable objects in Ku (Q) with the numerical class Pₗ, where Pₗ Ku (Q) is the projection sheaf of the skyscraper sheaf at a closed point x Q. We show that the moduli space Mₐ (v) of Gieseker semistable sheaves with Chern character v=ch (Pₗ) is smooth and irreducible of dimension four, and prove that the moduli space M () (Pₗ) is isomorphic to Mₐ (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 () (Pₗ). In the appendices, we show that the moduli space M () (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.