{"title":"Boundedness of regular del Pezzo surfaces over imperfect fields","authors":"Hiromu Tanaka","doi":"10.1007/s00229-023-01517-z","DOIUrl":null,"url":null,"abstract":"Abstract For a regular del Pezzo surface X , we prove that $$|-12K_X|$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mrow> <mml:mo>|</mml:mo> <mml:mo>-</mml:mo> <mml:mn>12</mml:mn> </mml:mrow> <mml:msub> <mml:mi>K</mml:mi> <mml:mi>X</mml:mi> </mml:msub> <mml:mrow> <mml:mo>|</mml:mo> </mml:mrow> </mml:mrow> </mml:math> is very ample. Furthermore, we also give an explicit upper bound for the volume $$K_X^2$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msubsup> <mml:mi>K</mml:mi> <mml:mi>X</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> </mml:math> which depends only on $$[k: k^p]$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mo>[</mml:mo> <mml:mi>k</mml:mi> <mml:mo>:</mml:mo> <mml:msup> <mml:mi>k</mml:mi> <mml:mi>p</mml:mi> </mml:msup> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> for the base field k . As a consequence, we obtain the boundedness of geometrically integral regular del Pezzo surfaces.","PeriodicalId":49887,"journal":{"name":"Manuscripta Mathematica","volume":"4 1","pages":"0"},"PeriodicalIF":0.5000,"publicationDate":"2023-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"5","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Manuscripta Mathematica","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s00229-023-01517-z","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: 5
Abstract
Abstract For a regular del Pezzo surface X , we prove that $$|-12K_X|$$ |-12KX| is very ample. Furthermore, we also give an explicit upper bound for the volume $$K_X^2$$ KX2 which depends only on $$[k: k^p]$$ [k:kp] for the base field k . As a consequence, we obtain the boundedness of geometrically integral regular del Pezzo surfaces.
期刊介绍：
manuscripta mathematica was founded in 1969 to provide a forum for the rapid communication of advances in mathematical research. Edited by an international board whose members represent a wide spectrum of research interests, manuscripta mathematica is now recognized as a leading source of information on the latest mathematical results.
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