关于连接性的几何判据及其在范数变体上的应用

IF 0.9 1区 数学 Q2 MATHEMATICS
Chetan T. Balwe, A. Hogadi, Anand Sawant
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Hogadi, Anand Sawant","doi":"10.1090/jag/790","DOIUrl":null,"url":null,"abstract":"<p>We show that <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper A Superscript 1\">\n <mml:semantics>\n <mml:msup>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">A</mml:mi>\n </mml:mrow>\n <mml:mn>1</mml:mn>\n </mml:msup>\n <mml:annotation encoding=\"application/x-tex\">\\mathbb {A}^1</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>-connectedness of a large class of varieties over a 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> can be characterized as the condition that their generic point can be connected to a <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>-rational point using (not necessarily naive) <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper A Superscript 1\">\n <mml:semantics>\n <mml:msup>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">A</mml:mi>\n </mml:mrow>\n <mml:mn>1</mml:mn>\n </mml:msup>\n <mml:annotation encoding=\"application/x-tex\">\\mathbb {A}^1</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>-homotopies. We also show that symmetric powers of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper A Superscript 1\">\n <mml:semantics>\n <mml:msup>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">A</mml:mi>\n </mml:mrow>\n <mml:mn>1</mml:mn>\n </mml:msup>\n <mml:annotation encoding=\"application/x-tex\">\\mathbb {A}^1</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>-connected smooth projective varieties (over an arbitrary field) as well as smooth proper models of them (over an algebraically closed field of characteristic <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"0\">\n <mml:semantics>\n <mml:mn>0</mml:mn>\n <mml:annotation encoding=\"application/x-tex\">0</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>) are <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper A Superscript 1\">\n <mml:semantics>\n <mml:msup>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">A</mml:mi>\n </mml:mrow>\n <mml:mn>1</mml:mn>\n </mml:msup>\n <mml:annotation encoding=\"application/x-tex\">\\mathbb {A}^1</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>-connected. As an application of these results, we show that the standard norm varieties over a 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> of characteristic <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"0\">\n <mml:semantics>\n <mml:mn>0</mml:mn>\n <mml:annotation encoding=\"application/x-tex\">0</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> become <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper A Superscript 1\">\n <mml:semantics>\n <mml:msup>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">A</mml:mi>\n </mml:mrow>\n <mml:mn>1</mml:mn>\n </mml:msup>\n <mml:annotation encoding=\"application/x-tex\">\\mathbb {A}^1</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>-connected (and consequently, universally <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper R\">\n <mml:semantics>\n <mml:mi>R</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">R</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>-trivial) after base change to an algebraic closure 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>.</p>","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2022-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"Geometric criteria for 𝔸¹-connectedness and applications to norm varieties\",\"authors\":\"Chetan T. 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引用次数: 1

摘要

我们证明了在域k k上的一大批变异的A 1 \mathbb {A}^1 -连通性可以表征为它们的一般点可以用(不一定是朴素的)A 1 \mathbb {A}^1 -同伦连接到k k -有理点的条件。我们还证明了A 1 \mathbb {A}^1连通光滑投影变型(在任意域上)的对称幂以及它们的光滑固有模型(在特征为0 0的代数闭域上)是A 1 \mathbb {A}^1连通的。作为这些结果的一个应用,我们证明了特征为0 0的域k k上的标准范数变体在基变换为k k的代数闭包后变成a1 \mathbb {a}^1 -连通(因此,普遍R R -平凡)。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Geometric criteria for 𝔸¹-connectedness and applications to norm varieties

We show that A 1 \mathbb {A}^1 -connectedness of a large class of varieties over a field k k can be characterized as the condition that their generic point can be connected to a k k -rational point using (not necessarily naive) A 1 \mathbb {A}^1 -homotopies. We also show that symmetric powers of A 1 \mathbb {A}^1 -connected smooth projective varieties (over an arbitrary field) as well as smooth proper models of them (over an algebraically closed field of characteristic 0 0 ) are A 1 \mathbb {A}^1 -connected. As an application of these results, we show that the standard norm varieties over a field k k of characteristic 0 0 become A 1 \mathbb {A}^1 -connected (and consequently, universally R R -trivial) after base change to an algebraic closure of k k .

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来源期刊
CiteScore
2.70
自引率
5.60%
发文量
23
审稿时长
>12 weeks
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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