The $d$-gonal locus in the moduli space of tropical plane curves
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| Format: | Preprint |
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2025
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| _version_ | 1866918218245865472 |
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| author | Leitz, Desmond Morrison, Ralph Newman-Taylor, Søren Wang, Vincent X. |
| author_facet | Leitz, Desmond Morrison, Ralph Newman-Taylor, Søren Wang, Vincent X. |
| contents | We introduce and study the locus $\mathbb{M}_{g,d}^\textrm{nd}$ of genus $g$ tropical plane curves of gonality $d$ inside the moduli space $\mathbb{M}^{\textrm{nd}}_{g}$ of tropical plane curves of genus $g$. Each such tropical curve arises from a Newton polygon, and we conjecture that the gonality of the tropical curve is equal to an easily computed parameter of this polygon called the expected gonality, closely related to the lattice width of the polygon. Let $\mathbb{M}_{g,{\underline{d}}}^\textrm{nd}$ denote the locus of tropical curves whose associated Newton polygon has expected gonality $d$. We prove that for fixed $d$ and sufficiently large genus $g$, the dimensions of these two loci agree: \[ \\dim\left(\mathbb{M}_{g,d}^\textrm{nd}\right) =\dim\left(\mathbb{M}_{g,{\underline{d}}}^\textrm{nd}\right). \] Our results provide evidence that, in sufficiently high genus compared to expected gonality, the gonality of a tropical curve is determined by the expected gonality of the Newton polygon from which it arises. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_20805 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The $d$-gonal locus in the moduli space of tropical plane curves Leitz, Desmond Morrison, Ralph Newman-Taylor, Søren Wang, Vincent X. Combinatorics Algebraic Geometry 14T05, 52B20, 14H10 We introduce and study the locus $\mathbb{M}_{g,d}^\textrm{nd}$ of genus $g$ tropical plane curves of gonality $d$ inside the moduli space $\mathbb{M}^{\textrm{nd}}_{g}$ of tropical plane curves of genus $g$. Each such tropical curve arises from a Newton polygon, and we conjecture that the gonality of the tropical curve is equal to an easily computed parameter of this polygon called the expected gonality, closely related to the lattice width of the polygon. Let $\mathbb{M}_{g,{\underline{d}}}^\textrm{nd}$ denote the locus of tropical curves whose associated Newton polygon has expected gonality $d$. We prove that for fixed $d$ and sufficiently large genus $g$, the dimensions of these two loci agree: \[ \\dim\left(\mathbb{M}_{g,d}^\textrm{nd}\right) =\dim\left(\mathbb{M}_{g,{\underline{d}}}^\textrm{nd}\right). \] Our results provide evidence that, in sufficiently high genus compared to expected gonality, the gonality of a tropical curve is determined by the expected gonality of the Newton polygon from which it arises. |
| title | The $d$-gonal locus in the moduli space of tropical plane curves |
| topic | Combinatorics Algebraic Geometry 14T05, 52B20, 14H10 |
| url | https://arxiv.org/abs/2511.20805 |