Secant rank and syzygies of projections of elliptic normal curves
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866910097491361792 |
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| author | Han, Changho Park, Euisung |
| author_facet | Han, Changho Park, Euisung |
| contents | We study the syzygies of projections of elliptic normal curves. Let $C \subset \mathbb{P}^{d-1}$ be an elliptic normal curve of degree $d \ge 5$, and let $C_q$ denote the projection of $C$ from a point $q$. We obtain sharp bounds for the Green--Lazarsfeld index of $C_q$ in terms of the secant rank of $q$. More precisely, if $q \in C^s \setminus C^2$, where $C^s$ is the $s$-th secant variety of $C$, then $\mathrm{index}(C_q) \le s-3$, and equality holds for a general point $q$ of $C^s$. In particular, $\mathrm{index}(C_q) = \lceil \frac{d}{2} \rceil - 3$ for a general point $q$ in $\mathbb{P}^{d-1}$. The proof realizes projected elliptic curves as hyperplane sections of elliptic ruled surface scrolls and exploits the known syzygetic properties of these scrolls. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_02046 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Secant rank and syzygies of projections of elliptic normal curves Han, Changho Park, Euisung Algebraic Geometry Primary: 14N15, Secondary: 51N35 We study the syzygies of projections of elliptic normal curves. Let $C \subset \mathbb{P}^{d-1}$ be an elliptic normal curve of degree $d \ge 5$, and let $C_q$ denote the projection of $C$ from a point $q$. We obtain sharp bounds for the Green--Lazarsfeld index of $C_q$ in terms of the secant rank of $q$. More precisely, if $q \in C^s \setminus C^2$, where $C^s$ is the $s$-th secant variety of $C$, then $\mathrm{index}(C_q) \le s-3$, and equality holds for a general point $q$ of $C^s$. In particular, $\mathrm{index}(C_q) = \lceil \frac{d}{2} \rceil - 3$ for a general point $q$ in $\mathbb{P}^{d-1}$. The proof realizes projected elliptic curves as hyperplane sections of elliptic ruled surface scrolls and exploits the known syzygetic properties of these scrolls. |
| title | Secant rank and syzygies of projections of elliptic normal curves |
| topic | Algebraic Geometry Primary: 14N15, Secondary: 51N35 |
| url | https://arxiv.org/abs/2604.02046 |