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| Main Author: | |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2302.06694 |
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| _version_ | 1866910639328329728 |
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| author | Vite, Montserrat |
| author_facet | Vite, Montserrat |
| contents | In the Hilbert scheme of curves of degree $d_{r}=\frac{r(r+1)}{2}$ and arithmetic genus $g_{r}=\frac{r(r+1)(2r-5)}{6}+1$ in $\mathbb{P}^{3}$ we prove that there exists a unique component of arithmetically Cohen-Macaulay curves denoted by $\overline{\mathscr{C}_{r}}$. For $r\geq 3$, we verify that the subvariety of curves in $\overline{\mathscr{C}_{r}}$ with Rao module of rank one always contains a reducible divisor. In particular, in the case of curves of degree $6$ and genus $3$ we prove that this subvariety is a reducible divisor. Furthermore, the components of such divisor are linearly independent and each component generates an extremal ray of the effective cone $\overline{\text{Eff}(\mathscr{C}_{3})}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_06694 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Liaison theory and the birational geometry of the Hilbert scheme of curves in $\mathbb{P}^{3}$ Vite, Montserrat Algebraic Geometry In the Hilbert scheme of curves of degree $d_{r}=\frac{r(r+1)}{2}$ and arithmetic genus $g_{r}=\frac{r(r+1)(2r-5)}{6}+1$ in $\mathbb{P}^{3}$ we prove that there exists a unique component of arithmetically Cohen-Macaulay curves denoted by $\overline{\mathscr{C}_{r}}$. For $r\geq 3$, we verify that the subvariety of curves in $\overline{\mathscr{C}_{r}}$ with Rao module of rank one always contains a reducible divisor. In particular, in the case of curves of degree $6$ and genus $3$ we prove that this subvariety is a reducible divisor. Furthermore, the components of such divisor are linearly independent and each component generates an extremal ray of the effective cone $\overline{\text{Eff}(\mathscr{C}_{3})}$. |
| title | Liaison theory and the birational geometry of the Hilbert scheme of curves in $\mathbb{P}^{3}$ |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2302.06694 |