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| Format: | Preprint |
| Published: |
2024
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| Online Access: | https://arxiv.org/abs/2412.10013 |
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| _version_ | 1866917875938230272 |
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| author | Yuan, Yao |
| author_facet | Yuan, Yao |
| contents | Let $X$ be a projective smooth surface over $\mathbb{C}$ with $H^2(\mathcal{O}_X)=0$. Let $M=M(L,χ)$ be the moduli space of 1-dimensional semistable sheaves with determinant $\mathcal{O}_X(L)$ and Euler characteristic $χ$. We have the Hilbert-Chow morphism $π:M\rightarrow |L|$. We give explicit forms of the higher direct images $R^iπ_*\mathcal{O}_M$ under some mild conditions on $M$ and $|L|$. Our result shows that $R^iπ_*\mathcal{O}_M$ are direct sums of line bundles. In particular, using our result one gets explicit formulas for the Euler characteristic of $π^*\mathcal{O}_{|L|}(m)$, which in $X=\mathbb{P}^2$ case was once conjectured by Chung-Moon. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_10013 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Higher direct images of the structure sheaf via the Hilbert-Chow morphism Yuan, Yao Algebraic Geometry 14D22, 14J26 Let $X$ be a projective smooth surface over $\mathbb{C}$ with $H^2(\mathcal{O}_X)=0$. Let $M=M(L,χ)$ be the moduli space of 1-dimensional semistable sheaves with determinant $\mathcal{O}_X(L)$ and Euler characteristic $χ$. We have the Hilbert-Chow morphism $π:M\rightarrow |L|$. We give explicit forms of the higher direct images $R^iπ_*\mathcal{O}_M$ under some mild conditions on $M$ and $|L|$. Our result shows that $R^iπ_*\mathcal{O}_M$ are direct sums of line bundles. In particular, using our result one gets explicit formulas for the Euler characteristic of $π^*\mathcal{O}_{|L|}(m)$, which in $X=\mathbb{P}^2$ case was once conjectured by Chung-Moon. |
| title | Higher direct images of the structure sheaf via the Hilbert-Chow morphism |
| topic | Algebraic Geometry 14D22, 14J26 |
| url | https://arxiv.org/abs/2412.10013 |