{"title":"The octanomial normal forms of cubic surfaces with applications to automorphisms","authors":"China Kaneko","doi":"10.1007/s10711-024-00931-1","DOIUrl":null,"url":null,"abstract":"<p>We will show that in any characteristic every nonsingular cubic surface is projectively isomorphic to the surface given by the octanomial normal form. This normal form is discovered in Panizzut et al. (LeMatematiche 75(2), 2020) only in characteristic 0 by exhaustive computer search. We offer a conceptual explanation that has the added benefit of being characteristic free. As an application, we give octanomial normal forms of the strata of the coarse moduli space of cubic surfaces defined in Dolgachev and Duncan (Compos Math 25(1):1–59, 1972) which preserve most specialization with respect to automorphisms.</p>","PeriodicalId":55103,"journal":{"name":"Geometriae Dedicata","volume":"43 1","pages":""},"PeriodicalIF":0.5000,"publicationDate":"2024-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Geometriae Dedicata","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10711-024-00931-1","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
We will show that in any characteristic every nonsingular cubic surface is projectively isomorphic to the surface given by the octanomial normal form. This normal form is discovered in Panizzut et al. (LeMatematiche 75(2), 2020) only in characteristic 0 by exhaustive computer search. We offer a conceptual explanation that has the added benefit of being characteristic free. As an application, we give octanomial normal forms of the strata of the coarse moduli space of cubic surfaces defined in Dolgachev and Duncan (Compos Math 25(1):1–59, 1972) which preserve most specialization with respect to automorphisms.
期刊介绍：
Geometriae Dedicata concentrates on geometry and its relationship to topology, group theory and the theory of dynamical systems.
Geometriae Dedicata aims to be a vehicle for excellent publications in geometry and related areas. Features of the journal will include:
A fast turn-around time for articles.
Special issues centered on specific topics.
All submitted papers should include some explanation of the context of the main results.
Geometriae Dedicata was founded in 1972 on the initiative of Hans Freudenthal in Utrecht, the Netherlands, who viewed geometry as a method rather than as a field. The present Board of Editors tries to continue in this spirit. The steady growth of the journal since its foundation is witness to the validity of the founder''s vision and to the success of the Editors'' mission.
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