Syzygies of canonical ribbons on higher genus curves
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916618775298048 |
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| author | Deopurkar, Anand Mukherjee, Jayan |
| author_facet | Deopurkar, Anand Mukherjee, Jayan |
| contents | We study the syzygies of the canonical embedding of a ribbon $\widetilde{C}$ on a curve $C$ of genus $g \geq 1$. We show that the linear series Clifford index and the resolution Clifford index are equal for a general ribbon of arithmetic genus $p_a$ on a general curve of genus $g$ with $p_{a} \geq \operatorname{max}\{3g+7, 6g-4\}$. Among non-general ribbons, the case of split ribbons is particularly interesting. Equality of the two Clifford indices for a split ribbon is related to the gonality conjecture for $C$ and it implies Green's conjecture for all double covers $C'$ of $C$ with $g(C') \geq \textrm{max}\{3g+2, 6g-4\}$. We reduce it to the vanishing of certain Koszul cohomology groups of an auxiliary module of syzygies associated to $C$, which may be of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_05500 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Syzygies of canonical ribbons on higher genus curves Deopurkar, Anand Mukherjee, Jayan Algebraic Geometry 14H51, 13D02 We study the syzygies of the canonical embedding of a ribbon $\widetilde{C}$ on a curve $C$ of genus $g \geq 1$. We show that the linear series Clifford index and the resolution Clifford index are equal for a general ribbon of arithmetic genus $p_a$ on a general curve of genus $g$ with $p_{a} \geq \operatorname{max}\{3g+7, 6g-4\}$. Among non-general ribbons, the case of split ribbons is particularly interesting. Equality of the two Clifford indices for a split ribbon is related to the gonality conjecture for $C$ and it implies Green's conjecture for all double covers $C'$ of $C$ with $g(C') \geq \textrm{max}\{3g+2, 6g-4\}$. We reduce it to the vanishing of certain Koszul cohomology groups of an auxiliary module of syzygies associated to $C$, which may be of independent interest. |
| title | Syzygies of canonical ribbons on higher genus curves |
| topic | Algebraic Geometry 14H51, 13D02 |
| url | https://arxiv.org/abs/2412.05500 |