Extension of Gorenstein weighted projective 3-spaces and characterization of the primitive curves of their surface sections
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866916593951309824 |
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| author | Dewer, Bruno |
| author_facet | Dewer, Bruno |
| contents | We investigate the Gorenstein weighted projective spaces of dimension 3. Given such a space $\mathbf P$, our first focus is its maximal extension in its anticanonical model $\mathbf P \subset \mathbf P^{g+1}$, i.e., the variety $Y\subset \mathbf P^{g+1+r}$ of largest dimension such that $Y$ is not a cone and $\mathbf P$ is a linear section of $Y$. In [DS23] Thomas Dedieu and Edoardo Sernesi have computed the dimension of $Y$ by cohomological computations on the canonical curves inside $\mathbf P$. We give an explicit description of $Y$ in the cases where it was not known. Next, we examine the general anticanonical divisors of $\mathbf P$. These are K3 surfaces, not necessarily primitively polarized. We give a geometric characterization of the curve sections in their primitive polarization. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2303_01882 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Extension of Gorenstein weighted projective 3-spaces and characterization of the primitive curves of their surface sections Dewer, Bruno Algebraic Geometry 14J30 (Primary) 14J28, 14H45 (Secondary) We investigate the Gorenstein weighted projective spaces of dimension 3. Given such a space $\mathbf P$, our first focus is its maximal extension in its anticanonical model $\mathbf P \subset \mathbf P^{g+1}$, i.e., the variety $Y\subset \mathbf P^{g+1+r}$ of largest dimension such that $Y$ is not a cone and $\mathbf P$ is a linear section of $Y$. In [DS23] Thomas Dedieu and Edoardo Sernesi have computed the dimension of $Y$ by cohomological computations on the canonical curves inside $\mathbf P$. We give an explicit description of $Y$ in the cases where it was not known. Next, we examine the general anticanonical divisors of $\mathbf P$. These are K3 surfaces, not necessarily primitively polarized. We give a geometric characterization of the curve sections in their primitive polarization. |
| title | Extension of Gorenstein weighted projective 3-spaces and characterization of the primitive curves of their surface sections |
| topic | Algebraic Geometry 14J30 (Primary) 14J28, 14H45 (Secondary) |
| url | https://arxiv.org/abs/2303.01882 |