Moduli of stable sheaves on quadric threefold
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866908358010732544 |
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| author | Yang, Song |
| author_facet | Yang, Song |
| contents | For each $0<α<\frac{1}{2}$, there exists a Bayer--Lahoz--Macr{\`ı}--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 $Q$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_18098 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Moduli of stable sheaves on quadric threefold Yang, Song Algebraic Geometry For each $0<α<\frac{1}{2}$, there exists a Bayer--Lahoz--Macr{\`ı}--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 $Q$. |
| title | Moduli of stable sheaves on quadric threefold |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2402.18098 |