与余维为1相交的线性子空间相关联的霍奇轨迹

IF 0.8 3区数学 Q2 MATHEMATICS

Mathematische Nachrichten Pub Date : 2025-02-28 DOI:10.1002/mana.202400066

Remke Kloosterman

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In this paper, we study the question whether the Hodge loci <math>\n <semantics>\n <mrow>\n <mo>NL</mo>\n <mo>(</mo>\n <mrow>\n <mo>[</mo>\n <msub>\n <mi>Π</mi>\n <mn>1</mn>\n </msub>\n <mo>]</mo>\n </mrow>\n <mo>+</mo>\n <mi>λ</mi>\n <mrow>\n <mo>[</mo>\n <msub>\n <mi>Π</mi>\n <mn>2</mn>\n </msub>\n <mo>]</mo>\n </mrow>\n <mo>)</mo>\n </mrow>\n <annotation>$\\operatorname{NL}([\\Pi _1]+\\lambda [\\Pi _2])$</annotation>\n </semantics></math> and <math>\n <semantics>\n <mrow>\n <mo>NL</mo>\n <mo>(</mo>\n <mrow>\n <mo>[</mo>\n <msub>\n <mi>Π</mi>\n <mn>1</mn>\n </msub>\n <mo>]</mo>\n </mrow>\n <mo>,</mo>\n <mrow>\n <mo>[</mo>\n <msub>\n <mi>Π</mi>\n <mn>2</mn>\n </msub>\n <mo>]</mo>\n </mrow>\n <mo>)</mo>\n </mrow>\n <annotation>$\\operatorname{NL}([\\Pi _1],[\\Pi _2])$</annotation>\n </semantics></math> coincide. This turns out to be the case in a neighborhood of <math>\n <semantics>\n <mi>X</mi>\n <annotation>$X$</annotation>\n </semantics></math> if <math>\n <semantics>\n <mi>X</mi>\n <annotation>$X$</annotation>\n </semantics></math> is very general on <math>\n <semantics>\n <mrow>\n <mo>NL</mo>\n <mo>(</mo>\n <mrow>\n <mo>[</mo>\n <msub>\n <mi>Π</mi>\n <mn>1</mn>\n </msub>\n <mo>]</mo>\n </mrow>\n <mo>,</mo>\n <mrow>\n <mo>[</mo>\n <msub>\n <mi>Π</mi>\n <mn>2</mn>\n </msub>\n <mo>]</mo>\n </mrow>\n <mo>)</mo>\n </mrow>\n <annotation>$\\operatorname{NL}([\\Pi _1],[\\Pi _2])$</annotation>\n </semantics></math>, <math>\n <semantics>\n <mrow>\n <mi>k</mi>\n <mo>></mo>\n <mn>1</mn>\n </mrow>\n <annotation>$k>1$</annotation>\n </semantics></math>, and <math>\n <semantics>\n <mrow>\n <mi>λ</mi>\n <mo>≠</mo>\n <mn>0</mn>\n <mo>,</mo>\n <mn>1</mn>\n </mrow>\n <annotation>$\\lambda \\ne 0,1$</annotation>\n </semantics></math>. However, there exists a hypersurface <math>\n <semantics>\n <mi>X</mi>\n <annotation>$X$</annotation>\n </semantics></math> for which <math>\n <semantics>\n <mrow>\n <mo>NL</mo>\n <mo>(</mo>\n <mrow>\n <mo>[</mo>\n <msub>\n <mi>Π</mi>\n <mn>1</mn>\n </msub>\n <mo>]</mo>\n </mrow>\n <mo>,</mo>\n <mrow>\n <mo>[</mo>\n <msub>\n <mi>Π</mi>\n <mn>2</mn>\n </msub>\n <mo>]</mo>\n </mrow>\n <mo>)</mo>\n </mrow>\n <annotation>$\\operatorname{NL}([\\Pi _1],[\\Pi _2])$</annotation>\n </semantics></math> is smooth at <math>\n <semantics>\n <mi>X</mi>\n <annotation>$X$</annotation>\n </semantics></math>, but <math>\n <semantics>\n <mrow>\n <mo>NL</mo>\n <mo>(</mo>\n <mrow>\n <mo>[</mo>\n <msub>\n <mi>Π</mi>\n <mn>1</mn>\n </msub>\n <mo>]</mo>\n </mrow>\n <mo>+</mo>\n <mi>λ</mi>\n <mrow>\n <mo>[</mo>\n <msub>\n <mi>Π</mi>\n <mn>2</mn>\n </msub>\n <mo>]</mo>\n </mrow>\n <mo>)</mo>\n </mrow>\n <annotation>$\\operatorname{NL}([\\Pi _1]+\\lambda [\\Pi _2])$</annotation>\n </semantics></math> is singular for all <math>\n <semantics>\n <mrow>\n <mi>λ</mi>\n <mo>≠</mo>\n <mn>0</mn>\n <mo>,</mo>\n <mn>1</mn>\n </mrow>\n <annotation>$\\lambda \\ne 0,1$</annotation>\n </semantics></math>. We expect that this is due to an embedded component of <math>\n <semantics>\n <mrow>\n <mo>NL</mo>\n <mo>(</mo>\n <mrow>\n <mo>[</mo>\n <msub>\n <mi>Π</mi>\n <mn>1</mn>\n </msub>\n <mo>]</mo>\n </mrow>\n <mo>+</mo>\n <mi>λ</mi>\n <mrow>\n <mo>[</mo>\n <msub>\n <mi>Π</mi>\n <mn>2</mn>\n </msub>\n <mo>]</mo>\n </mrow>\n <mo>)</mo>\n </mrow>\n <annotation>$\\operatorname{NL}([\\Pi _1]+\\lambda [\\Pi _2])$</annotation>\n </semantics></math>. The case <math>\n <semantics>\n <mrow>\n <mi>k</mi>\n <mo>=</mo>\n <mn>1</mn>\n </mrow>\n <annotation>$k=1$</annotation>\n </semantics></math> was treated before by Dan, in that case <math>\n <semantics>\n <mrow>\n <mo>NL</mo>\n <mo>(</mo>\n <mrow>\n <mo>[</mo>\n <msub>\n <mi>Π</mi>\n <mn>1</mn>\n </msub>\n <mo>]</mo>\n </mrow>\n <mo>+</mo>\n <mi>λ</mi>\n <mrow>\n <mo>[</mo>\n <msub>\n <mi>Π</mi>\n <mn>2</mn>\n </msub>\n <mo>]</mo>\n </mrow>\n <mo>)</mo>\n </mrow>\n <annotation>$\\operatorname{NL}([\\Pi _1]+\\lambda [\\Pi _2])$</annotation>\n </semantics></math> is nonreduced.","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"298 4","pages":"1220-1229"},"PeriodicalIF":0.8000,"publicationDate":"2025-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/mana.202400066","citationCount":"0","resultStr":"{\"title\":\"Hodge loci associated with linear subspaces intersecting in codimension one\",\"authors\":\"Remke Kloosterman\",\"doi\":\"10.1002/mana.202400066\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let <math>\\n <semantics>\\n <mrow>\\n <mi>X</mi>\\n <mo>⊂</mo>\\n <msup>\\n <mi>P</mi>\\n <mrow>\\n <mn>2</mn>\\n <mi>k</mi>\\n <mo>+</mo>\\n <mn>1</mn>\\n </mrow>\\n </msup>\\n </mrow>\\n <annotation>$X\\\\subset \\\\mathbf {P}^{2k+1}$</annotation>\\n </semantics></math> be a smooth hypersurface containing two <math>\\n <semantics>\\n <mi>k</mi>\\n <annotation>$k$</annotation>\\n </semantics></math>-dimensional linear spaces <math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>Π</mi>\\n <mn>1</mn>\\n </msub>\\n <mo>,</mo>\\n <msub>\\n <mi>Π</mi>\\n <mn>2</mn>\\n </msub>\\n </mrow>\\n <annotation>$\\\\Pi _1,\\\\Pi _2$</annotation>\\n </semantics></math>, such that <math>\\n <semantics>\\n <mrow>\\n <mo>dim</mo>\\n <msub>\\n <mi>Π</mi>\\n <mn>1</mn>\\n </msub>\\n <mo>∩</mo>\\n <msub>\\n <mi>Π</mi>\\n <mn>2</mn>\\n </msub>\\n <mo>=</mo>\\n <mi>k</mi>\\n <mo>−</mo>\\n <mn>1</mn>\\n </mrow>\\n <annotation>$\\\\dim \\\\Pi _1\\\\cap \\\\Pi _2=k-1$</annotation>\\n </semantics></math>. In this paper, we study the question whether the Hodge loci <math>\\n <semantics>\\n <mrow>\\n <mo>NL</mo>\\n <mo>(</mo>\\n <mrow>\\n <mo>[</mo>\\n <msub>\\n <mi>Π</mi>\\n <mn>1</mn>\\n </msub>\\n <mo>]</mo>\\n </mrow>\\n <mo>+</mo>\\n <mi>λ</mi>\\n <mrow>\\n <mo>[</mo>\\n <msub>\\n <mi>Π</mi>\\n <mn>2</mn>\\n </msub>\\n <mo>]</mo>\\n </mrow>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$\\\\operatorname{NL}([\\\\Pi _1]+\\\\lambda [\\\\Pi _2])$</annotation>\\n </semantics></math> and <math>\\n <semantics>\\n <mrow>\\n <mo>NL</mo>\\n <mo>(</mo>\\n <mrow>\\n <mo>[</mo>\\n <msub>\\n <mi>Π</mi>\\n <mn>1</mn>\\n </msub>\\n <mo>]</mo>\\n </mrow>\\n <mo>,</mo>\\n <mrow>\\n <mo>[</mo>\\n <msub>\\n <mi>Π</mi>\\n <mn>2</mn>\\n </msub>\\n <mo>]</mo>\\n </mrow>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$\\\\operatorname{NL}([\\\\Pi _1],[\\\\Pi _2])$</annotation>\\n </semantics></math> coincide. This turns out to be the case in a neighborhood of <math>\\n <semantics>\\n <mi>X</mi>\\n <annotation>$X$</annotation>\\n </semantics></math> if <math>\\n <semantics>\\n <mi>X</mi>\\n <annotation>$X$</annotation>\\n </semantics></math> is very general on <math>\\n <semantics>\\n <mrow>\\n <mo>NL</mo>\\n <mo>(</mo>\\n <mrow>\\n <mo>[</mo>\\n <msub>\\n <mi>Π</mi>\\n <mn>1</mn>\\n </msub>\\n <mo>]</mo>\\n </mrow>\\n <mo>,</mo>\\n <mrow>\\n <mo>[</mo>\\n <msub>\\n <mi>Π</mi>\\n <mn>2</mn>\\n </msub>\\n <mo>]</mo>\\n </mrow>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$\\\\operatorname{NL}([\\\\Pi _1],[\\\\Pi _2])$</annotation>\\n </semantics></math>, <math>\\n <semantics>\\n <mrow>\\n <mi>k</mi>\\n <mo>></mo>\\n <mn>1</mn>\\n </mrow>\\n <annotation>$k>1$</annotation>\\n </semantics></math>, and <math>\\n <semantics>\\n <mrow>\\n <mi>λ</mi>\\n <mo>≠</mo>\\n <mn>0</mn>\\n <mo>,</mo>\\n <mn>1</mn>\\n </mrow>\\n <annotation>$\\\\lambda \\\\ne 0,1$</annotation>\\n </semantics></math>. However, there exists a hypersurface <math>\\n <semantics>\\n <mi>X</mi>\\n <annotation>$X$</annotation>\\n </semantics></math> for which <math>\\n <semantics>\\n <mrow>\\n <mo>NL</mo>\\n <mo>(</mo>\\n <mrow>\\n <mo>[</mo>\\n <msub>\\n <mi>Π</mi>\\n <mn>1</mn>\\n </msub>\\n <mo>]</mo>\\n </mrow>\\n <mo>,</mo>\\n <mrow>\\n <mo>[</mo>\\n <msub>\\n <mi>Π</mi>\\n <mn>2</mn>\\n </msub>\\n <mo>]</mo>\\n </mrow>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$\\\\operatorname{NL}([\\\\Pi _1],[\\\\Pi _2])$</annotation>\\n </semantics></math> is smooth at <math>\\n <semantics>\\n <mi>X</mi>\\n <annotation>$X$</annotation>\\n </semantics></math>, but <math>\\n <semantics>\\n <mrow>\\n <mo>NL</mo>\\n <mo>(</mo>\\n <mrow>\\n <mo>[</mo>\\n <msub>\\n <mi>Π</mi>\\n <mn>1</mn>\\n </msub>\\n <mo>]</mo>\\n </mrow>\\n <mo>+</mo>\\n <mi>λ</mi>\\n <mrow>\\n <mo>[</mo>\\n <msub>\\n <mi>Π</mi>\\n <mn>2</mn>\\n </msub>\\n <mo>]</mo>\\n </mrow>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$\\\\operatorname{NL}([\\\\Pi _1]+\\\\lambda [\\\\Pi _2])$</annotation>\\n </semantics></math> is singular for all <math>\\n <semantics>\\n <mrow>\\n <mi>λ</mi>\\n <mo>≠</mo>\\n <mn>0</mn>\\n <mo>,</mo>\\n <mn>1</mn>\\n </mrow>\\n <annotation>$\\\\lambda \\\\ne 0,1$</annotation>\\n </semantics></math>. We expect that this is due to an embedded component of <math>\\n <semantics>\\n <mrow>\\n <mo>NL</mo>\\n <mo>(</mo>\\n <mrow>\\n <mo>[</mo>\\n <msub>\\n <mi>Π</mi>\\n <mn>1</mn>\\n </msub>\\n <mo>]</mo>\\n </mrow>\\n <mo>+</mo>\\n <mi>λ</mi>\\n <mrow>\\n <mo>[</mo>\\n <msub>\\n <mi>Π</mi>\\n <mn>2</mn>\\n </msub>\\n <mo>]</mo>\\n </mrow>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$\\\\operatorname{NL}([\\\\Pi _1]+\\\\lambda [\\\\Pi _2])$</annotation>\\n </semantics></math>. The case <math>\\n <semantics>\\n <mrow>\\n <mi>k</mi>\\n <mo>=</mo>\\n <mn>1</mn>\\n </mrow>\\n <annotation>$k=1$</annotation>\\n </semantics></math> was treated before by Dan, in that case <math>\\n <semantics>\\n <mrow>\\n <mo>NL</mo>\\n <mo>(</mo>\\n <mrow>\\n <mo>[</mo>\\n <msub>\\n <mi>Π</mi>\\n <mn>1</mn>\\n </msub>\\n <mo>]</mo>\\n </mrow>\\n <mo>+</mo>\\n <mi>λ</mi>\\n <mrow>\\n <mo>[</mo>\\n <msub>\\n <mi>Π</mi>\\n <mn>2</mn>\\n </msub>\\n <mo>]</mo>\\n </mrow>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$\\\\operatorname{NL}([\\\\Pi _1]+\\\\lambda [\\\\Pi _2])$</annotation>\\n </semantics></math> is nonreduced.\",\"PeriodicalId\":49853,\"journal\":{\"name\":\"Mathematische Nachrichten\",\"volume\":\"298 4\",\"pages\":\"1220-1229\"},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2025-02-28\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://onlinelibrary.wiley.com/doi/epdf/10.1002/mana.202400066\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Mathematische Nachrichten\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1002/mana.202400066\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematische Nachrichten","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/mana.202400066","RegionNum":3,"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

摘要

设X∧p2 k + 1 $X\subset \mathbf {P}^{2k+1}$是一个包含2k的光滑超曲面$k$-维线性空间Π 1， Π 2 $\Pi _1,\Pi _2$，使dim Π 1∩Π 2 = k−1 $\dim \Pi _1\cap \Pi _2=k-1$。在本文中，我们研究了Hodge基因座NL ([Π 1] + λ [Π 2 .]) $\operatorname{NL}([\Pi _1]+\lambda [\Pi _2])$和NL ([Π 1]，[Π 2]) $\operatorname{NL}([\Pi _1],[\Pi _2])$重合。如果X $X$在NL （[Π 1]）上是非常一般的，那么在X $X$的邻域内就是这种情况， [Π 2]) $\operatorname{NL}([\Pi _1],[\Pi _2])$, k ＆gt；1 $k>1$, λ≠0,1 $\lambda \ne 0,1$。然而，存在一个超曲面X $X$，其中NL ([Π 1]，[Π 2]) $\operatorname{NL}([\Pi _1],[\Pi _2])$在X $X$平滑，但NL ([Π 1] + λ [Π 2]) $\operatorname{NL}([\Pi _1]+\lambda [\Pi _2])$对于所有λ≠0,1 $\lambda \ne 0,1$都是奇异的。我们预计这是由于NL ([Π 1] + λ [Π 2]的嵌入式组件]) $\operatorname{NL}([\Pi _1]+\lambda [\Pi _2])$。病例k = 1 $k=1$之前由Dan处理过，在这种情况下NL ([Π 1] + λ [Π 2]]) $\operatorname{NL}([\Pi _1]+\lambda [\Pi _2])$是非还原的。

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Hodge loci associated with linear subspaces intersecting in codimension one

Let $X \subset P^{2 k + 1}$ be a smooth hypersurface containing two $k$ -dimensional linear spaces $Π_{1}, Π_{2}$ , such that $dim Π_{1} \cap Π_{2} = k - 1$ . In this paper, we study the question whether the Hodge loci $NL ([Π_{1}] + λ [Π_{2}])$ and $NL ([Π_{1}], [Π_{2}])$ coincide. This turns out to be the case in a neighborhood of $X$ if $X$ is very general on $NL ([Π_{1}], [Π_{2}])$ , $k > 1$ , and $λ \neq 0, 1$ . However, there exists a hypersurface $X$ for which $NL ([Π_{1}], [Π_{2}])$ is smooth at $X$ , but $NL ([Π_{1}] + λ [Π_{2}])$ is singular for all $λ \neq 0, 1$ . We expect that this is due to an embedded component of $NL ([Π_{1}] + λ [Π_{2}])$ . The case $k = 1$ was treated before by Dan, in that case $NL ([Π_{1}] + λ [Π_{2}])$ is nonreduced.

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来源期刊

Mathematische Nachrichten 数学-数学

CiteScore

1.50

自引率

0.00%

发文量

157

审稿时长

4-8 weeks

期刊介绍： Mathematische Nachrichten - Mathematical News publishes original papers on new results and methods that hold prospect for substantial progress in mathematics and its applications. All branches of analysis, algebra, number theory, geometry and topology, flow mechanics and theoretical aspects of stochastics are given special emphasis. Mathematische Nachrichten is indexed/abstracted in Current Contents/Physical, Chemical and Earth Sciences; Mathematical Review; Zentralblatt für Mathematik; Math Database on STN International, INSPEC; Science Citation Index