Moving curves of least gonality on symmetric products of curves
| Autor: | Bastianelli, Francesco, Picoco, Nicola |
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| Rok vydání: | 2024 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: The paper is a sequel of arXiv:2208.00990v3, and the main result was originally included in arXiv:2208.00990v2. v1: 20 pages - v2: 21 pages; the proof of the main result has been reorganized |
| Databáze: | arXiv |
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