Hilbert scheme and Hilbert functions of smooth curves of degrees at most $15$ in $\mathbb{P}^5$
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| Format: | Preprint |
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2025
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| _version_ | 1866912496769564672 |
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| author | Ballico, Edoardo Keem, Changho |
| author_facet | Ballico, Edoardo Keem, Changho |
| contents | Denoting $\mathcal{H}_{d,g,5}$ by the Hilbert scheme of smooth curves of degree $d$ and genus $g$ in $\mathbb{P}^5$, let $\mathcal{H}$ be an irreducible component of $\mathcal{H}_{d,g,5}$. We study the Hilbert function $h_X:\mathbb{N}\longrightarrow\mathbb{N}$, $h_X(t):= h^0(\mathcal{I}_X(t))$ of a general member $X\in\mathcal{H}$, especially when the degree of the curve is low; $d\le 15$. We also determine the irreducibility of $\mathcal{H}_{d,g,5}$ for $d\le 14$ and study the natural functorial map $μ:$\mathcal{H}_{d,g,5}$ \longrightarrow \mathcal{M}_g$ in some detail. We describe the fibre $μ^{-1}μ(X)$ for a general $X\in\mathcal{H} $ as well as determining the projective normality (or being ACM). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_16547 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hilbert scheme and Hilbert functions of smooth curves of degrees at most $15$ in $\mathbb{P}^5$ Ballico, Edoardo Keem, Changho Algebraic Geometry Primary 14C05, Secondary 14H10 Denoting $\mathcal{H}_{d,g,5}$ by the Hilbert scheme of smooth curves of degree $d$ and genus $g$ in $\mathbb{P}^5$, let $\mathcal{H}$ be an irreducible component of $\mathcal{H}_{d,g,5}$. We study the Hilbert function $h_X:\mathbb{N}\longrightarrow\mathbb{N}$, $h_X(t):= h^0(\mathcal{I}_X(t))$ of a general member $X\in\mathcal{H}$, especially when the degree of the curve is low; $d\le 15$. We also determine the irreducibility of $\mathcal{H}_{d,g,5}$ for $d\le 14$ and study the natural functorial map $μ:$\mathcal{H}_{d,g,5}$ \longrightarrow \mathcal{M}_g$ in some detail. We describe the fibre $μ^{-1}μ(X)$ for a general $X\in\mathcal{H} $ as well as determining the projective normality (or being ACM). |
| title | Hilbert scheme and Hilbert functions of smooth curves of degrees at most $15$ in $\mathbb{P}^5$ |
| topic | Algebraic Geometry Primary 14C05, Secondary 14H10 |
| url | https://arxiv.org/abs/2507.16547 |