Moduli of stable sheaves on quadric threefold

For each 0<α<1/2, there exists a Bayer--Lahoz--Macr{\`{}}--Stellari's inducing Bridgeland stability condition σ(α) on a Kuznetsov component Ku(Q) of the smooth quadric threefold Q. We obtain the non-empty of the moduli space Mσ(α)([Pₓ]) of σ(α)-semistable objects in Ku(Q) with the numerical class [Pₓ], where Pₓ∈ Ku(Q) is the projection sheaf of the skyscraper sheaf at a closed point x∈ Q. We show that the moduli space M̄Q(v) of Gieseker semistable sheaves with Chern character v=ch(Pₓ) is smooth and irreducible of dimension four, and prove that the moduli space Mσ(α)([Pₓ]) is isomorphic to M̄Q(v). As an application, we show that the quadric threefold Q can be reinterpreted as a Brill--Noether locus in the Bridgeland moduli space Mσ(α)([Pₓ]). In the appendices, we show that the moduli space Mσ(α)([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.