Moving curves of least gonality on symmetric products of curves
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929321491300352 |
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| author | Bastianelli, Francesco Picoco, Nicola |
| author_facet | Bastianelli, Francesco Picoco, Nicola |
| contents | This paper is a sequel of arXiv:2208.00990. Let $C$ be a smooth complex projective curve of genus $g$ and let $C^{(k)}$ be its $k$-fold symmetric product. The covering gonality of $C^{(k)}$ is the least gonality of an irreducible curve $E\subset C^{(k)}$ passing through a general point of $C^{(k)}$. It follows from previous works of the authors that if $2\leq k\leq 4$ and $g\geq k+4$, the covering gonality of $C^{(k)}$ equals the gonality of $C$. In this paper, we prove that under mild assumptions of generality on $C$, the only curves $E\subset C^{(k)}$ computing the covering gonality of $C^{(k)}$ are copies of $C$ of the form $C+p$, for some point $p\in C^{(k-1)}$. As a byproduct, we deduce that the connecting gonality of $C^{(k)}$ (i.e. the least gonality of an irreducible curve $E\subset C^{(k)}$ connecting two general points of $C^{(k)}$) is strictly larger than the covering gonality. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_00753 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Moving curves of least gonality on symmetric products of curves Bastianelli, Francesco Picoco, Nicola Algebraic Geometry This paper is a sequel of arXiv:2208.00990. Let $C$ be a smooth complex projective curve of genus $g$ and let $C^{(k)}$ be its $k$-fold symmetric product. The covering gonality of $C^{(k)}$ is the least gonality of an irreducible curve $E\subset C^{(k)}$ passing through a general point of $C^{(k)}$. It follows from previous works of the authors that if $2\leq k\leq 4$ and $g\geq k+4$, the covering gonality of $C^{(k)}$ equals the gonality of $C$. In this paper, we prove that under mild assumptions of generality on $C$, the only curves $E\subset C^{(k)}$ computing the covering gonality of $C^{(k)}$ are copies of $C$ of the form $C+p$, for some point $p\in C^{(k-1)}$. As a byproduct, we deduce that the connecting gonality of $C^{(k)}$ (i.e. the least gonality of an irreducible curve $E\subset C^{(k)}$ connecting two general points of $C^{(k)}$) is strictly larger than the covering gonality. |
| title | Moving curves of least gonality on symmetric products of curves |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2402.00753 |