Hilbert scheme of linearly normal curves in $\mathbb{P}^r$ with index of speciality five and beyond
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866913225973432320 |
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| author | Keem, Changho |
| author_facet | Keem, Changho |
| contents | We study the Hilbert scheme of smooth, irreducible, non-degenerate and linearly normal curves of degree $d$ and genus $g$ in $\mathbb{P}^r$ ($r\ge 3$) whose complete and very ample hyperplane linear series $\mathcal{D}$ have relatively small index of speciality $i(\mathcal{D})=g-d+r$. In particular we determine the existence as well as the non-existence of Hilbert schemes of linearly normal curves $\mathcal{H}^{\mathcal{L}}_{d,g,r}$ for every possible triples $(d,g,r)$ with $i(\mathcal{D})=5$ and $r\ge 3$. We also determine the irreducibility of the Hilbert scheme $\mathcal{H}^{\mathcal{L}}_{g+r-5,g,r}$ when the genus $g$ is near to the minimal possible value with respect to the dimension of the projective space $\mathbb{P}^r$ for which $\mathcal{H}^{\mathcal{L}}_{g+r-5,g,r}\neq\emptyset$, say $r+9\le g\le r+11$. In the course of proofs of key results, we show the existence of linearly normal curves of degree $d\ge g+1$ with arbitrarily given index of speciality with some mild restriction on the genus $g$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_01347 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Hilbert scheme of linearly normal curves in $\mathbb{P}^r$ with index of speciality five and beyond Keem, Changho Algebraic Geometry Primary 14C05, Secondary 14H10 We study the Hilbert scheme of smooth, irreducible, non-degenerate and linearly normal curves of degree $d$ and genus $g$ in $\mathbb{P}^r$ ($r\ge 3$) whose complete and very ample hyperplane linear series $\mathcal{D}$ have relatively small index of speciality $i(\mathcal{D})=g-d+r$. In particular we determine the existence as well as the non-existence of Hilbert schemes of linearly normal curves $\mathcal{H}^{\mathcal{L}}_{d,g,r}$ for every possible triples $(d,g,r)$ with $i(\mathcal{D})=5$ and $r\ge 3$. We also determine the irreducibility of the Hilbert scheme $\mathcal{H}^{\mathcal{L}}_{g+r-5,g,r}$ when the genus $g$ is near to the minimal possible value with respect to the dimension of the projective space $\mathbb{P}^r$ for which $\mathcal{H}^{\mathcal{L}}_{g+r-5,g,r}\neq\emptyset$, say $r+9\le g\le r+11$. In the course of proofs of key results, we show the existence of linearly normal curves of degree $d\ge g+1$ with arbitrarily given index of speciality with some mild restriction on the genus $g$. |
| title | Hilbert scheme of linearly normal curves in $\mathbb{P}^r$ with index of speciality five and beyond |
| topic | Algebraic Geometry Primary 14C05, Secondary 14H10 |
| url | https://arxiv.org/abs/2312.01347 |