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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2406.01051 |
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| _version_ | 1866910469337382912 |
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| author | Haghighi, Hassan Mosakhani, Mohammad |
| author_facet | Haghighi, Hassan Mosakhani, Mohammad |
| contents | The purpose of this paper is to construct some special kind of subschemes in $\mathbb{P}^N$ with $ N\ge 3$, which we call them "fat flat subschemes" and compute their Waldschmidt constants. These subschemes are constructed by adding, in a particular way, a finite number of linear subspaces of $\mathbb{P}^N$ of many different dimensions to a star configuration in $\mathbb{P}^N$, with arbitrary preassigned multiplicities to each one of these linear subspaces, as well as the star configuration. Among other things, it will be shown that for every positive integer $d$, there are infinitely many fat flat subschemes in $\mathbb{P}^N$ with the Waldschmidt constant equal to $d$. In addition to this, for any two integers $1\le a<b$, we also construct a fat flat subscheme of the above type in some projective space $\mathbb{P}^M$, which its Waldschmidt constant is equal to $b/a$. In addition to these, all non-reduced fat points subschemes $Z$ in $\mathbb{P}^2$ with the Waldschmidt constants less than $5/2$ are classified. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_01051 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Waldschmidt constant of special fat flat subschemes in $\mathbb{P}^N$]{The Waldschmidt constant of special fat flat subschemes in $\mathbb{P}^N$ Haghighi, Hassan Mosakhani, Mohammad Algebraic Geometry Primary 14N20, 13A02, Secondary 14N05, 13F20 The purpose of this paper is to construct some special kind of subschemes in $\mathbb{P}^N$ with $ N\ge 3$, which we call them "fat flat subschemes" and compute their Waldschmidt constants. These subschemes are constructed by adding, in a particular way, a finite number of linear subspaces of $\mathbb{P}^N$ of many different dimensions to a star configuration in $\mathbb{P}^N$, with arbitrary preassigned multiplicities to each one of these linear subspaces, as well as the star configuration. Among other things, it will be shown that for every positive integer $d$, there are infinitely many fat flat subschemes in $\mathbb{P}^N$ with the Waldschmidt constant equal to $d$. In addition to this, for any two integers $1\le a<b$, we also construct a fat flat subscheme of the above type in some projective space $\mathbb{P}^M$, which its Waldschmidt constant is equal to $b/a$. In addition to these, all non-reduced fat points subschemes $Z$ in $\mathbb{P}^2$ with the Waldschmidt constants less than $5/2$ are classified. |
| title | The Waldschmidt constant of special fat flat subschemes in $\mathbb{P}^N$]{The Waldschmidt constant of special fat flat subschemes in $\mathbb{P}^N$ |
| topic | Algebraic Geometry Primary 14N20, 13A02, Secondary 14N05, 13F20 |
| url | https://arxiv.org/abs/2406.01051 |