Crossing the transcendental divide: from Schottky groups to algebraic curves
Fuente:
arXiv
Guardado en:
| Autores principales: | , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2024
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866916105319088128 |
|---|---|
| author | Fairchild, Samantha Ortiz, Ángel David Ríos |
| author_facet | Fairchild, Samantha Ortiz, Ángel David Ríos |
| contents | Though the uniformization theorem guarantees an equivalence of Riemann surfaces and smooth algebraic curves, moving between analytic and algebraic representations is inherently transcendental. Our analytic curves identify pairs of circles in the complex plane via free groups of Möbius transformations called Schottky groups. We construct a family of non-hyperelliptic surfaces of genus $g\geq 3$ where we know the Riemann surface as well as properties of the canonical embedding, including a nontrivial symmetry group and a real structure with the maximal number of connected components (an $M$-curve). We then numerically approximate the algebraic curve and Riemann matrices underlying our family of Riemann surfaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_10801 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Crossing the transcendental divide: from Schottky groups to algebraic curves Fairchild, Samantha Ortiz, Ángel David Ríos Geometric Topology Algebraic Geometry 30F40, 14Q05, 20H10, 22E40 Though the uniformization theorem guarantees an equivalence of Riemann surfaces and smooth algebraic curves, moving between analytic and algebraic representations is inherently transcendental. Our analytic curves identify pairs of circles in the complex plane via free groups of Möbius transformations called Schottky groups. We construct a family of non-hyperelliptic surfaces of genus $g\geq 3$ where we know the Riemann surface as well as properties of the canonical embedding, including a nontrivial symmetry group and a real structure with the maximal number of connected components (an $M$-curve). We then numerically approximate the algebraic curve and Riemann matrices underlying our family of Riemann surfaces. |
| title | Crossing the transcendental divide: from Schottky groups to algebraic curves |
| topic | Geometric Topology Algebraic Geometry 30F40, 14Q05, 20H10, 22E40 |
| url | https://arxiv.org/abs/2401.10801 |