{"title":"Invariants of vanishing Brauer classes","authors":"Federica Galluzzi, Bert van Geemen","doi":"10.1007/s40687-024-00459-6","DOIUrl":null,"url":null,"abstract":"<p>A specialization of a <i>K</i>3 surface with Picard rank one to a <i>K</i>3 with rank two defines a vanishing class of order two in the Brauer group of the general <i>K</i>3 surface. We give the <i>B</i>-field invariants of this class. We apply this to the <i>K</i>3 double plane defined by a cubic fourfold with a plane. The specialization of such a cubic fourfold whose group of codimension two cycles has rank two to one which has rank three induces such a specialization of the double planes. We determine the Picard lattice of the specialized double plane as well as the vanishing Brauer class and its relation to the natural ‘Clifford’ Brauer class. This provides more insight in the specializations. It allows us to explicitly determine the <i>K</i>3 surfaces associated with infinitely many of the conjecturally rational cubic fourfolds obtained as such specializations.</p>","PeriodicalId":48561,"journal":{"name":"Research in the Mathematical Sciences","volume":"45 1","pages":""},"PeriodicalIF":1.2000,"publicationDate":"2024-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Research in the Mathematical Sciences","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s40687-024-00459-6","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
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Abstract
A specialization of a K3 surface with Picard rank one to a K3 with rank two defines a vanishing class of order two in the Brauer group of the general K3 surface. We give the B-field invariants of this class. We apply this to the K3 double plane defined by a cubic fourfold with a plane. The specialization of such a cubic fourfold whose group of codimension two cycles has rank two to one which has rank three induces such a specialization of the double planes. We determine the Picard lattice of the specialized double plane as well as the vanishing Brauer class and its relation to the natural ‘Clifford’ Brauer class. This provides more insight in the specializations. It allows us to explicitly determine the K3 surfaces associated with infinitely many of the conjecturally rational cubic fourfolds obtained as such specializations.
期刊介绍：
Research in the Mathematical Sciences is an international, peer-reviewed hybrid journal covering the full scope of Theoretical Mathematics, Applied Mathematics, and Theoretical Computer Science. The Mission of the Journal is to publish high-quality original articles that make a significant contribution to the research areas of both theoretical and applied mathematics and theoretical computer science.
This journal is an efficient enterprise where the editors play a central role in soliciting the best research papers, and where editorial decisions are reached in a timely fashion. Research in the Mathematical Sciences does not have a length restriction and encourages the submission of longer articles in which more complex and detailed analysis and proofing of theorems is required. It also publishes shorter research communications (Letters) covering nascent research in some of the hottest areas of mathematical research. This journal will publish the highest quality papers in all of the traditional areas of applied and theoretical areas of mathematics and computer science, and it will actively seek to publish seminal papers in the most emerging and interdisciplinary areas in all of the mathematical sciences. Research in the Mathematical Sciences wishes to lead the way by promoting the highest quality research of this type.
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