Moduli spaces of sheaves on Fano threefolds and K3 surfaces of genus 9
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arXiv
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| Formato: | Preprint |
| Publicado: |
2020
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| _version_ | 1866917558461923328 |
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| author | Mattei, Dominique |
| author_facet | Mattei, Dominique |
| contents | A complex smooth prime Fano threefold $X$ of genus $9$ is related via projective duality to a quartic plane curve $Γ$. We use this setup to study the restriction of rank $2$ stable sheaves with prescribed Chern classes on $X$ to an anticanonical $K3$ surface $S\subset X$. Varying the threefold $X$ containing $S$ gives a rational Lagrangian fibration $$\mathcal{M}_S(2,1,7) \dashrightarrow \mathbb{P}^3$$ with generic fibre birational to the moduli space $\mathcal{M}_X(2,1,7)$ of sheaves on $X$. Moreover, we prove that this rational fibration extends to an actual fibration on a birational model $\mathcal{M}$ of $\mathcal{M}_S(2,1,7)$.
In a last part, we use Bridgeland stability conditions to exhibit all $K$-trivial smooth birational models of $\mathcal{M}_S(2,1,7)$, which consist in itself and $\mathcal{M}$. We prove that these models are related by a flop, and we describe the positive, movable and nef cones of $\mathcal{M}_S(2,1,7)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2012_05848 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Moduli spaces of sheaves on Fano threefolds and K3 surfaces of genus 9 Mattei, Dominique Algebraic Geometry 14D20, 14J60, 14D06, 14F08 A complex smooth prime Fano threefold $X$ of genus $9$ is related via projective duality to a quartic plane curve $Γ$. We use this setup to study the restriction of rank $2$ stable sheaves with prescribed Chern classes on $X$ to an anticanonical $K3$ surface $S\subset X$. Varying the threefold $X$ containing $S$ gives a rational Lagrangian fibration $$\mathcal{M}_S(2,1,7) \dashrightarrow \mathbb{P}^3$$ with generic fibre birational to the moduli space $\mathcal{M}_X(2,1,7)$ of sheaves on $X$. Moreover, we prove that this rational fibration extends to an actual fibration on a birational model $\mathcal{M}$ of $\mathcal{M}_S(2,1,7)$. In a last part, we use Bridgeland stability conditions to exhibit all $K$-trivial smooth birational models of $\mathcal{M}_S(2,1,7)$, which consist in itself and $\mathcal{M}$. We prove that these models are related by a flop, and we describe the positive, movable and nef cones of $\mathcal{M}_S(2,1,7)$. |
| title | Moduli spaces of sheaves on Fano threefolds and K3 surfaces of genus 9 |
| topic | Algebraic Geometry 14D20, 14J60, 14D06, 14F08 |
| url | https://arxiv.org/abs/2012.05848 |