The moduli space of Harnack curves in toric surfaces
In 2006, Kenyon and Okounkov Kenyon and Okounkov [12] computed the moduli space of Harnack curves of degree d in ${\mathbb {C}\mathbb {P}}^2$. We generalise their construction to any projective toric surface and show that the moduli space ${\mathcal {H}_\Delta }$ of Harnack curves with Newton polygo...
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Cambridge University Press
2021-01-01
|
| Series: | Forum of Mathematics, Sigma |
| Subjects: | |
| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509421000372/type/journal_article |
| _version_ | 1811156198131499008 |
|---|---|
| author | Jorge Alberto Olarte |
| author_facet | Jorge Alberto Olarte |
| author_sort | Jorge Alberto Olarte |
| collection | DOAJ |
| description | In 2006, Kenyon and Okounkov Kenyon and Okounkov [12] computed the moduli space of Harnack curves of degree d in ${\mathbb {C}\mathbb {P}}^2$. We generalise their construction to any projective toric surface and show that the moduli space ${\mathcal {H}_\Delta }$ of Harnack curves with Newton polygon $\Delta $ is diffeomorphic to ${\mathbb {R}}^{m-3}\times {\mathbb {R}}_{\geq 0}^{n+g-m}$, where $\Delta $ has m edges, g interior lattice points and n boundary lattice points. This solves a conjecture of Crétois and Lang. The main result uses abstract tropical curves to construct a compactification of this moduli space where additional points correspond to collections of curves that can be patchworked together to produce a curve in ${\mathcal {H}_\Delta }$. This compactification has a natural stratification with the same poset as the secondary polytope of $\Delta $. |
| first_indexed | 2024-04-10T04:47:29Z |
| format | Article |
| id | doaj.art-bc99e9a9eff84ed892b41eb617eab064 |
| institution | Directory Open Access Journal |
| issn | 2050-5094 |
| language | English |
| last_indexed | 2024-04-10T04:47:29Z |
| publishDate | 2021-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Sigma |
| spelling | doaj.art-bc99e9a9eff84ed892b41eb617eab0642023-03-09T12:34:52ZengCambridge University PressForum of Mathematics, Sigma2050-50942021-01-01910.1017/fms.2021.37The moduli space of Harnack curves in toric surfacesJorge Alberto Olarte0https://orcid.org/0000-0002-1280-1265Institut für Mathematik, Technische Universität Berlin, Straße des 17. Juni 135, Berlin, Germany; E-mail:In 2006, Kenyon and Okounkov Kenyon and Okounkov [12] computed the moduli space of Harnack curves of degree d in ${\mathbb {C}\mathbb {P}}^2$. We generalise their construction to any projective toric surface and show that the moduli space ${\mathcal {H}_\Delta }$ of Harnack curves with Newton polygon $\Delta $ is diffeomorphic to ${\mathbb {R}}^{m-3}\times {\mathbb {R}}_{\geq 0}^{n+g-m}$, where $\Delta $ has m edges, g interior lattice points and n boundary lattice points. This solves a conjecture of Crétois and Lang. The main result uses abstract tropical curves to construct a compactification of this moduli space where additional points correspond to collections of curves that can be patchworked together to produce a curve in ${\mathcal {H}_\Delta }$. This compactification has a natural stratification with the same poset as the secondary polytope of $\Delta $.https://www.cambridge.org/core/product/identifier/S2050509421000372/type/journal_article14H5014H1014M2552B2014T20 |
| spellingShingle | Jorge Alberto Olarte The moduli space of Harnack curves in toric surfaces Forum of Mathematics, Sigma 14H50 14H10 14M25 52B20 14T20 |
| title | The moduli space of Harnack curves in toric surfaces |
| title_full | The moduli space of Harnack curves in toric surfaces |
| title_fullStr | The moduli space of Harnack curves in toric surfaces |
| title_full_unstemmed | The moduli space of Harnack curves in toric surfaces |
| title_short | The moduli space of Harnack curves in toric surfaces |
| title_sort | moduli space of harnack curves in toric surfaces |
| topic | 14H50 14H10 14M25 52B20 14T20 |
| url | https://www.cambridge.org/core/product/identifier/S2050509421000372/type/journal_article |
| work_keys_str_mv | AT jorgealbertoolarte themodulispaceofharnackcurvesintoricsurfaces AT jorgealbertoolarte modulispaceofharnackcurvesintoricsurfaces |