Non-reduced components of the Hilbert scheme of curves using triple covers
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866916250685276160 |
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| author | Choi, Youngook Iliev, Hristo Kim, Seonja |
| author_facet | Choi, Youngook Iliev, Hristo Kim, Seonja |
| contents | In this paper we consider curves on a cone that pass through the vertex and are also triple covers of the base of the cone, which is a general smooth curve of genus $γ$ and degree $e$ in $\mathbb{P}^{e-γ}$. Using the free resolution of the ideal of such a curve found by Catalisano and Gimigliano, and a technique concerning deformations of curves introduced by Ciliberto, we show that the deformations of such curves remain on cones over a deformation of the base curve. This allows us to prove that for $γ\geq 3$ and $e \geq 4γ+ 5$ there exists a non-reduced component $\mathcal{H}$ of the Hilbert scheme of smooth curves of genus $3e + 3γ$ and degree $3e+1$ in $\mathbb{P}^{e-γ+1}$. We show that $\dim T_{[X]} \mathcal{H} = \dim \mathcal{H} + 1 = (e - γ+ 1)^2 + 7e + 5$ for a general point $[X] \in \mathcal{H}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_08707 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Non-reduced components of the Hilbert scheme of curves using triple covers Choi, Youngook Iliev, Hristo Kim, Seonja Algebraic Geometry 14C05, 14H10 In this paper we consider curves on a cone that pass through the vertex and are also triple covers of the base of the cone, which is a general smooth curve of genus $γ$ and degree $e$ in $\mathbb{P}^{e-γ}$. Using the free resolution of the ideal of such a curve found by Catalisano and Gimigliano, and a technique concerning deformations of curves introduced by Ciliberto, we show that the deformations of such curves remain on cones over a deformation of the base curve. This allows us to prove that for $γ\geq 3$ and $e \geq 4γ+ 5$ there exists a non-reduced component $\mathcal{H}$ of the Hilbert scheme of smooth curves of genus $3e + 3γ$ and degree $3e+1$ in $\mathbb{P}^{e-γ+1}$. We show that $\dim T_{[X]} \mathcal{H} = \dim \mathcal{H} + 1 = (e - γ+ 1)^2 + 7e + 5$ for a general point $[X] \in \mathcal{H}$. |
| title | Non-reduced components of the Hilbert scheme of curves using triple covers |
| topic | Algebraic Geometry 14C05, 14H10 |
| url | https://arxiv.org/abs/2302.08707 |