{"title":"Codimension two integral points on some rationally connected threefolds are potentially dense","authors":"David McKinnon, Mike Roth","doi":"10.1090/jag/782","DOIUrl":null,"url":null,"abstract":"<p>Let <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper X\">\n <mml:semantics>\n <mml:mi>X</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">X</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> be a smooth, projective, rationally connected variety, defined over a number field <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"k\">\n <mml:semantics>\n <mml:mi>k</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">k</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, and let <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper Z subset-of upper X\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>Z</mml:mi>\n <mml:mo>⊂<!-- ⊂ --></mml:mo>\n <mml:mi>X</mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">Z\\subset X</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> be a closed subset of codimension at least two. In this paper, for certain choices of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper X\">\n <mml:semantics>\n <mml:mi>X</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">X</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, we prove that the set of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper Z\">\n <mml:semantics>\n <mml:mi>Z</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">Z</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>-integral points is potentially Zariski dense, in the sense that there is a finite extension <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K\">\n <mml:semantics>\n <mml:mi>K</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">K</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"k\">\n <mml:semantics>\n <mml:mi>k</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">k</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> such that the set of points <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper P element-of upper X left-parenthesis upper K right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>P</mml:mi>\n <mml:mo>∈<!-- ∈ --></mml:mo>\n <mml:mi>X</mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>K</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">P\\in X(K)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> that are <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper Z\">\n <mml:semantics>\n <mml:mi>Z</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">Z</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>-integral is Zariski dense in <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper X\">\n <mml:semantics>\n <mml:mi>X</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">X</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. This gives a positive answer to a question of Hassett and Tschinkel from 2001.</p>","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2020-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Algebraic Geometry","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/jag/782","RegionNum":1,"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
Let XX be a smooth, projective, rationally connected variety, defined over a number field kk, and let Z⊂XZ\subset X be a closed subset of codimension at least two. In this paper, for certain choices of XX, we prove that the set of ZZ-integral points is potentially Zariski dense, in the sense that there is a finite extension KK of kk such that the set of points P∈X(K)P\in X(K) that are ZZ-integral is Zariski dense in XX. This gives a positive answer to a question of Hassett and Tschinkel from 2001.
期刊介绍：
The Journal of Algebraic Geometry is devoted to research articles in algebraic geometry, singularity theory, and related subjects such as number theory, commutative algebra, projective geometry, complex geometry, and geometric topology.
This journal, published quarterly with articles electronically published individually before appearing in an issue, is distributed by the American Mathematical Society (AMS). In order to take advantage of some features offered for this journal, users will occasionally be linked to pages on the AMS website.
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