On the local cohomology of secant varieties
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913442603991040 |
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| author | Olano, Sebastian Raychaudhury, Debaditya |
| author_facet | Olano, Sebastian Raychaudhury, Debaditya |
| contents | Given a sufficiently positive embedding $X\subset\mathbb{P}^N$ of a smooth projective variety $X$, we consider its secant variety $Σ$ that comes equipped with the embedding $Σ\subset\mathbb{P}^N$ by its construction. In this article, we determine the local cohomological dimension $\textrm{lcd}(\mathbb{P}^N,Σ)$ of this embedding, as well as the generation level of the Hodge filtration on the topmost non-vanishing local cohomology module $\mathcal{H}^{q}_Σ(\mathcal{O}_{\mathbb{P}^N})$, i.e., when $q=\textrm{lcd}(\mathbb{P}^N,Σ)$. Additionally, we show that $Σ$ has quotient singularities (in which case the equality $\textrm{lcd}(\mathbb{P}^N,Σ)=\textrm{codim}_{\mathbb{P}^N}(Σ)$ is known to hold) if and only if $X\cong\mathbb{P}^1$. We also provide a complete classification of $(X,L)$ for which $Σ$ has ($\mathbb{Q}$-)Gorentein singularities. As a consequence, we deduce that if $Σ$ is a local complete intersection, then either $X$ is isomorphic to $\mathbb{P}^1$, or an elliptic curve. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_16688 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the local cohomology of secant varieties Olano, Sebastian Raychaudhury, Debaditya Algebraic Geometry Given a sufficiently positive embedding $X\subset\mathbb{P}^N$ of a smooth projective variety $X$, we consider its secant variety $Σ$ that comes equipped with the embedding $Σ\subset\mathbb{P}^N$ by its construction. In this article, we determine the local cohomological dimension $\textrm{lcd}(\mathbb{P}^N,Σ)$ of this embedding, as well as the generation level of the Hodge filtration on the topmost non-vanishing local cohomology module $\mathcal{H}^{q}_Σ(\mathcal{O}_{\mathbb{P}^N})$, i.e., when $q=\textrm{lcd}(\mathbb{P}^N,Σ)$. Additionally, we show that $Σ$ has quotient singularities (in which case the equality $\textrm{lcd}(\mathbb{P}^N,Σ)=\textrm{codim}_{\mathbb{P}^N}(Σ)$ is known to hold) if and only if $X\cong\mathbb{P}^1$. We also provide a complete classification of $(X,L)$ for which $Σ$ has ($\mathbb{Q}$-)Gorentein singularities. As a consequence, we deduce that if $Σ$ is a local complete intersection, then either $X$ is isomorphic to $\mathbb{P}^1$, or an elliptic curve. |
| title | On the local cohomology of secant varieties |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2407.16688 |