Moduli of stable sheaves on quadric threefold

TL;DR

The work constructs Bridgeland stability conditions on the Kuznetsov component $ ext{Ku}(Q)$ of a smooth quadric threefold $Q$ via the BLMS inducing framework and analyzes moduli of stable objects with the primitive class $2oldsymbol{}_{2}-oldsymbol{}_{1}$. It proves the non-emptiness of the Bridgeland moduli $M_{\sigma}(2oldsymbol{}_{2}-oldsymbol{}_{1})$ for stability conditions in the relevant orbit, and shows this moduli is isomorphic to the Gieseker moduli $ar M_Q( extbf{v})$ with $ extbf{v}= ext{ch}( ext{P}_x)=(3,-H,- frac12 H^{2}, frac13 H^{3})$, which is smooth and irreducible of dimension $4$. As an application, the quadric threefold $Q$ is realized as a Brill–Noether locus inside the Bridgeland moduli, linking geometric points to Ext-dimension conditions with a universal object; appendices address the unique stability of the spinor bundle and the Hilbert scheme of lines. The results connect Kuznetsov-component stability, classical Gieseker theory, and Brill–Noether geometry in a coherent framework, expanding the understanding of moduli in rigid Fano geometries.

Abstract

For each $0<α<\frac{1}{2}$, there exists a Bayer--Lahoz--Macr{ì}--Stellari inducing Bridgeland stability condition $σ(α)$ on a Kuznetsov component $\mathrm{Ku}(Q)$ of the smooth quadric threefold $Q$. We obtain the non-emptiness of the moduli space $M_{σ(α)}([\mathcal{P}_{x}])$ of $σ(α)$-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 $M_{σ(α)}([\mathcal{P}_{x}])$ 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 $M_{σ(α)}([\mathcal{P}_{x}])$. In the appendices, we show that the moduli space $M_{σ(α)}([S])$ contains only one single point corresponding to the spinor bundle $S$ and give a Bridgeland moduli interpretation for the Hilbert scheme of lines in $Q$.

Theorems & Definitions (96)

• Proposition 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 2.1
• Definition 2.2
• Remark 2.3
• Definition 2.4
• Definition 2.5
• Definition 2.6
• Proposition 2.7: HRS96