Constructions of superabundant tropical curves in higher genus

IF 0.8 3区数学 Q2 MATHEMATICS

Bulletin of the London Mathematical Society Pub Date : 2024-12-14 DOI:10.1112/blms.13209

Sae Koyama

{"title":"Constructions of superabundant tropical curves in higher genus","authors":"Sae Koyama","doi":"10.1112/blms.13209","DOIUrl":null,"url":null,"abstract":"We construct qualitatively new examples of superabundant tropical curves which are non-realisable in genuses 3 and 4. These curves are in <math>\n <semantics>\n <msup>\n <mi>R</mi>\n <mn>3</mn>\n </msup>\n <annotation>${\\mathbb {R}}^3$</annotation>\n </semantics></math> and <math>\n <semantics>\n <msup>\n <mi>R</mi>\n <mn>4</mn>\n </msup>\n <annotation>${\\mathbb {R}}^4$</annotation>\n </semantics></math>, respectively, and have properties resembling canonical embeddings of genus 3 and 4 algebraic curves. In particular, the genus 3 example is a degree 4 planar tropical curve, and the genus 4 example is contained in the product of a tropical line and a tropical conic. They have excess dimension of deformation space equal to 1. Non-realisability follows by combining this with a dimension calculation for the corresponding space of logarithmic curves.","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 2","pages":"490-509"},"PeriodicalIF":0.8000,"publicationDate":"2024-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/blms.13209","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Bulletin of the London Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/blms.13209","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 0

Abstract

We construct qualitatively new examples of superabundant tropical curves which are non-realisable in genuses 3 and 4. These curves are in $R^{3}$ and $R^{4}$ , respectively, and have properties resembling canonical embeddings of genus 3 and 4 algebraic curves. In particular, the genus 3 example is a degree 4 planar tropical curve, and the genus 4 example is contained in the product of a tropical line and a tropical conic. They have excess dimension of deformation space equal to 1. Non-realisability follows by combining this with a dimension calculation for the corresponding space of logarithmic curves.