Belyi Function Decompositions for The Icosahedron of Genus 4
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| Format: | Preprint |
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2024
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| _version_ | 1866914793372254208 |
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| author | Amburg, Natalia Kovaleva, Mariia |
| author_facet | Amburg, Natalia Kovaleva, Mariia |
| contents | The icosahedron $I_4$ of genus 4 is a dessin d'enfant embedded in Bring's curve $\mathcal{B}$. The dessin $I_4$ is related in some sense to a regular icosahedron $I_0$ embedded in the complex Riemann sphere. In particular, decompositions of Belyi functions $β_{I_0}: \mathbb{CP}^1 \rightarrow \mathbb{CP}^1$ and $β_{I_4}: \mathcal{B} \rightarrow \mathbb{CP}^1$ for $I_0$ and $I_4$ have the same lattice. The diagram of $β_{I_0}$ decompositions is already known. In the present paper we find $β_{I_4}$ decompositions. Note that $β_{I_0}$ decomposes into rational functions on $\mathbb{C}P^1$, while in case of $β_{I_4}$ we deal with maps between different algebraic curves. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_07092 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Belyi Function Decompositions for The Icosahedron of Genus 4 Amburg, Natalia Kovaleva, Mariia Algebraic Geometry 14H57, 14H45 The icosahedron $I_4$ of genus 4 is a dessin d'enfant embedded in Bring's curve $\mathcal{B}$. The dessin $I_4$ is related in some sense to a regular icosahedron $I_0$ embedded in the complex Riemann sphere. In particular, decompositions of Belyi functions $β_{I_0}: \mathbb{CP}^1 \rightarrow \mathbb{CP}^1$ and $β_{I_4}: \mathcal{B} \rightarrow \mathbb{CP}^1$ for $I_0$ and $I_4$ have the same lattice. The diagram of $β_{I_0}$ decompositions is already known. In the present paper we find $β_{I_4}$ decompositions. Note that $β_{I_0}$ decomposes into rational functions on $\mathbb{C}P^1$, while in case of $β_{I_4}$ we deal with maps between different algebraic curves. |
| title | Belyi Function Decompositions for The Icosahedron of Genus 4 |
| topic | Algebraic Geometry 14H57, 14H45 |
| url | https://arxiv.org/abs/2405.07092 |