{"title":"Hyperelliptic genus 3 curves with involutions and a Prym map","authors":"Paweł Borówka, Anatoli Shatsila","doi":"10.1002/mana.202300468","DOIUrl":null,"url":null,"abstract":"<p>We characterize genus 3 complex smooth hyperelliptic curves that admit two additional involutions as curves that can be built from five points in <span></span><math>\n <semantics>\n <msup>\n <mi>P</mi>\n <mn>1</mn>\n </msup>\n <annotation>$\\mathbb {P}^1$</annotation>\n </semantics></math> with a distinguished triple. We are able to write down explicit equations for the curves and all their quotient curves. We show that, fixing one of the elliptic quotient curve, the Prym map becomes a 2:1 map and therefore the hyperelliptic Klein Prym map, constructed recently by the first author with A. Ortega, is also 2:1 in this case. As a by-product we show an explicit family of <span></span><math>\n <semantics>\n <mrow>\n <mo>(</mo>\n <mn>1</mn>\n <mo>,</mo>\n <mi>d</mi>\n <mo>)</mo>\n </mrow>\n <annotation>$(1,d)$</annotation>\n </semantics></math> polarized abelian surfaces (for <span></span><math>\n <semantics>\n <mrow>\n <mi>d</mi>\n <mo>></mo>\n <mn>1</mn>\n </mrow>\n <annotation>$d&gt;1$</annotation>\n </semantics></math>), such that any surface in the family satisfying a certain explicit condition is abstractly non-isomorphic to its dual abelian surface.</p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"297 8","pages":"3080-3094"},"PeriodicalIF":0.8000,"publicationDate":"2024-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematische Nachrichten","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/mana.202300468","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 characterize genus 3 complex smooth hyperelliptic curves that admit two additional involutions as curves that can be built from five points in with a distinguished triple. We are able to write down explicit equations for the curves and all their quotient curves. We show that, fixing one of the elliptic quotient curve, the Prym map becomes a 2:1 map and therefore the hyperelliptic Klein Prym map, constructed recently by the first author with A. Ortega, is also 2:1 in this case. As a by-product we show an explicit family of polarized abelian surfaces (for ), such that any surface in the family satisfying a certain explicit condition is abstractly non-isomorphic to its dual abelian surface.
期刊介绍:
Mathematische Nachrichten - Mathematical News publishes original papers on new results and methods that hold prospect for substantial progress in mathematics and its applications. All branches of analysis, algebra, number theory, geometry and topology, flow mechanics and theoretical aspects of stochastics are given special emphasis. Mathematische Nachrichten is indexed/abstracted in Current Contents/Physical, Chemical and Earth Sciences; Mathematical Review; Zentralblatt für Mathematik; Math Database on STN International, INSPEC; Science Citation Index