On the rank index of projective curves of almost minimal degree
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866908880484696064 |
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| author | Jung, Jaewoo Moon, Hyunsuk Park, Euisung |
| author_facet | Jung, Jaewoo Moon, Hyunsuk Park, Euisung |
| contents | In this article, we investigate the rank index of projective curves $\mathscr{C} \subset \mathbb{P}^r$ of degree $r+1$ when $\mathscr{C} = π_p (\tilde{\mathscr{C}})$ for the standard rational normal curve $\tilde{\mathscr{C}} \subset \mathbb{P}^{r+1}$ and a point $p \in \mathbb{P}^{r+1} \setminus \tilde{\mathscr{C}}^3$. Here, the rank index of a closed subscheme $X \subset \mathbb{P}^r$ is defined to be the least integer $k$ such that its homogeneous ideal can be generated by quadratic polynomials of rank $\leq k$. Our results show that the rank index of $\mathscr{C}$ is at most $4$, and it is exactly equal to $3$ when the projection center $p$ is a coordinate point of $\mathbb{P}^{r+1}$. We also investigate the case where $p \in \tilde{\mathscr{C}}^3 \setminus \tilde{\mathscr{C}}^2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_17494 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the rank index of projective curves of almost minimal degree Jung, Jaewoo Moon, Hyunsuk Park, Euisung Algebraic Geometry 14A25, 14H45, 14N05, 15A63, 16E45 In this article, we investigate the rank index of projective curves $\mathscr{C} \subset \mathbb{P}^r$ of degree $r+1$ when $\mathscr{C} = π_p (\tilde{\mathscr{C}})$ for the standard rational normal curve $\tilde{\mathscr{C}} \subset \mathbb{P}^{r+1}$ and a point $p \in \mathbb{P}^{r+1} \setminus \tilde{\mathscr{C}}^3$. Here, the rank index of a closed subscheme $X \subset \mathbb{P}^r$ is defined to be the least integer $k$ such that its homogeneous ideal can be generated by quadratic polynomials of rank $\leq k$. Our results show that the rank index of $\mathscr{C}$ is at most $4$, and it is exactly equal to $3$ when the projection center $p$ is a coordinate point of $\mathbb{P}^{r+1}$. We also investigate the case where $p \in \tilde{\mathscr{C}}^3 \setminus \tilde{\mathscr{C}}^2$. |
| title | On the rank index of projective curves of almost minimal degree |
| topic | Algebraic Geometry 14A25, 14H45, 14N05, 15A63, 16E45 |
| url | https://arxiv.org/abs/2411.17494 |