On the rank index of projective curves of almost minimal degree
| Autor: | Jung, Jaewoo, Moon, Hyunsuk, Park, Euisung |
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| Rok vydání: | 2024 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | In this article, we investigate the rank index of projective curves $\mathscr{C} \subset \mathbb{P}^r$ of degree $r+1$ when $\mathscr{C} = \pi_p (\tilde{\mathscr{C}})$ 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 $p \in \tilde{\mathscr{C}}^3 \setminus \tilde{\mathscr{C}}^2$. Comment: 24 pages |
| Databáze: | arXiv |
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