Moduli spaces of one dimensional sheaves on log surfaces and Hilbert schemes
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916715776966656 |
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| author | Takahashi, Nobuyoshi |
| author_facet | Takahashi, Nobuyoshi |
| contents | Let $X$ be a smooth projective rational surface, $D\subset X$ an effective anticanonical curve, $β$ a curve class on $X$ and $\mathfrak{d}=\sum w_iP_i$ an effective divisor on $D_{\mathrm{sm}}$. We consider the moduli space $\mathcal{M}_β(X, D, \mathfrak{d})$ of sheaves on $X$ which are direct images of rank-$1$ torsion-free sheaves on integral curves $C$ in $β$ such that $C|_D=\mathfrak{d}$, and show that each point of $\mathcal{M}_β(X, D, \mathfrak{d})$ is smooth over a point in the product of the Hilbert schemes of surface singularities of types $A_{w_i-1}$. Hence, $\mathcal{M}_β(X, D, \mathfrak{d})$ has symplectic singularities and admits a unique symplectic resolution. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_00246 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Moduli spaces of one dimensional sheaves on log surfaces and Hilbert schemes Takahashi, Nobuyoshi Algebraic Geometry Primary 14B05, Secondary 14H40, 14H60, 14J42 Let $X$ be a smooth projective rational surface, $D\subset X$ an effective anticanonical curve, $β$ a curve class on $X$ and $\mathfrak{d}=\sum w_iP_i$ an effective divisor on $D_{\mathrm{sm}}$. We consider the moduli space $\mathcal{M}_β(X, D, \mathfrak{d})$ of sheaves on $X$ which are direct images of rank-$1$ torsion-free sheaves on integral curves $C$ in $β$ such that $C|_D=\mathfrak{d}$, and show that each point of $\mathcal{M}_β(X, D, \mathfrak{d})$ is smooth over a point in the product of the Hilbert schemes of surface singularities of types $A_{w_i-1}$. Hence, $\mathcal{M}_β(X, D, \mathfrak{d})$ has symplectic singularities and admits a unique symplectic resolution. |
| title | Moduli spaces of one dimensional sheaves on log surfaces and Hilbert schemes |
| topic | Algebraic Geometry Primary 14B05, Secondary 14H40, 14H60, 14J42 |
| url | https://arxiv.org/abs/2505.00246 |