Components of the Hilbert Scheme of smooth projective curves using ruled surfaces II: existence of non-reduced components
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| Format: | Preprint |
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2022
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| _version_ | 1866917706297507840 |
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| author | Choi, Youngook Iliev, Hristo Kim, Seonja |
| author_facet | Choi, Youngook Iliev, Hristo Kim, Seonja |
| contents | For $γ\geq 7$ and $g \geq 6γ+ 5$, we construct a family $\mathcal{F}^{\prime}$ of curves lying on cones in $\mathbb{P}^{g-3γ+1}$ over smooth non-degenerate curves of genus $γ$ and degree $g-2γ$ in $\mathbb{P}^{g-3γ+1}$. We show that $\dim \mathcal{F}^{\prime} = 2g-γ-1 + (g-3γ+1)^2$. For a general curve $X^{\prime}$ from the family $\mathcal{F}^{\prime}$, we compute the dimension of the space of its first-order deformations. We prove that the family $\mathcal{F}^{\prime}$ gives rise to an irreducible, non-reduced component $\mathcal{D}^{\prime}$ of the Hilbert scheme $\mathcal{I}_{2g-4γ+ 1, g, g - 3γ+ 1}$, which parametrizes smooth, irreducible, non-degenerate curves of degree $2g-4γ+ 1$ and genus $g$ in $\mathbb{P}^{g-3γ+1}$. We obtain $\dim T_{[X^{\prime}]} \mathcal{D}^{\prime} = \dim \mathcal{D}^{\prime} + 1 = \dim \mathcal{F}^{\prime} + 1$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2208_12470 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Components of the Hilbert Scheme of smooth projective curves using ruled surfaces II: existence of non-reduced components Choi, Youngook Iliev, Hristo Kim, Seonja Algebraic Geometry 14C05, 14H10 For $γ\geq 7$ and $g \geq 6γ+ 5$, we construct a family $\mathcal{F}^{\prime}$ of curves lying on cones in $\mathbb{P}^{g-3γ+1}$ over smooth non-degenerate curves of genus $γ$ and degree $g-2γ$ in $\mathbb{P}^{g-3γ+1}$. We show that $\dim \mathcal{F}^{\prime} = 2g-γ-1 + (g-3γ+1)^2$. For a general curve $X^{\prime}$ from the family $\mathcal{F}^{\prime}$, we compute the dimension of the space of its first-order deformations. We prove that the family $\mathcal{F}^{\prime}$ gives rise to an irreducible, non-reduced component $\mathcal{D}^{\prime}$ of the Hilbert scheme $\mathcal{I}_{2g-4γ+ 1, g, g - 3γ+ 1}$, which parametrizes smooth, irreducible, non-degenerate curves of degree $2g-4γ+ 1$ and genus $g$ in $\mathbb{P}^{g-3γ+1}$. We obtain $\dim T_{[X^{\prime}]} \mathcal{D}^{\prime} = \dim \mathcal{D}^{\prime} + 1 = \dim \mathcal{F}^{\prime} + 1$. |
| title | Components of the Hilbert Scheme of smooth projective curves using ruled surfaces II: existence of non-reduced components |
| topic | Algebraic Geometry 14C05, 14H10 |
| url | https://arxiv.org/abs/2208.12470 |