{"title":"On varieties whose general surface section has negative Kodaira dimension","authors":"Ciro Ciliberto, Claudio Fontanari","doi":"10.1002/mana.202300565","DOIUrl":null,"url":null,"abstract":"<p>In this paper, inspired by work of Fano, Morin, and Campana–Flenner, we give a projective classification of varieties of dimension 3 whose general hyperplane sections have negative Kodaira dimension, and we partly extend such a classification to varieties of dimension <span></span><math>\n <semantics>\n <mrow>\n <mi>n</mi>\n <mo>⩾</mo>\n <mn>4</mn>\n </mrow>\n <annotation>$n\\geqslant 4$</annotation>\n </semantics></math> whose general surface sections have negative Kodaira dimension. In particular, we prove that a variety of dimension <span></span><math>\n <semantics>\n <mrow>\n <mi>n</mi>\n <mo>⩾</mo>\n <mn>3</mn>\n </mrow>\n <annotation>$n\\geqslant 3$</annotation>\n </semantics></math> whose general surface sections have negative Kodaira dimension is birationally equivalent to the product of a general surface section times <span></span><math>\n <semantics>\n <msup>\n <mi>P</mi>\n <mrow>\n <mi>n</mi>\n <mo>−</mo>\n <mn>2</mn>\n </mrow>\n </msup>\n <annotation>${\\mathbb {P}}^{n-2}$</annotation>\n </semantics></math> unless (possibly) if the variety is a cubic hypersurface.</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/mana.202300565","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
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Abstract
In this paper, inspired by work of Fano, Morin, and Campana–Flenner, we give a projective classification of varieties of dimension 3 whose general hyperplane sections have negative Kodaira dimension, and we partly extend such a classification to varieties of dimension whose general surface sections have negative Kodaira dimension. In particular, we prove that a variety of dimension whose general surface sections have negative Kodaira dimension is birationally equivalent to the product of a general surface section times unless (possibly) if the variety is a cubic hypersurface.