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{"title":"关于Veronese嵌入的3阶二次方程","authors":"Euisung Park, Saerom Sim","doi":"10.1002/mana.70028","DOIUrl":null,"url":null,"abstract":"<p>This paper studies the geometric structure of the locus <span></span><math>\n <semantics>\n <mrow>\n <msub>\n <mi>Φ</mi>\n <mn>3</mn>\n </msub>\n <mrow>\n <mo>(</mo>\n <mi>X</mi>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation>$\\Phi _3 (X)$</annotation>\n </semantics></math> of rank 3 quadratic equations of the Veronese variety <span></span><math>\n <semantics>\n <mrow>\n <mi>X</mi>\n <mo>=</mo>\n <msub>\n <mi>ν</mi>\n <mi>d</mi>\n </msub>\n <mrow>\n <mo>(</mo>\n <msup>\n <mi>P</mi>\n <mi>n</mi>\n </msup>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation>$X = \\nu _d ({\\mathbb {P}}^n)$</annotation>\n </semantics></math>. Specifically, we investigate the minimal irreducible decomposition of <span></span><math>\n <semantics>\n <mrow>\n <msub>\n <mi>Φ</mi>\n <mn>3</mn>\n </msub>\n <mrow>\n <mo>(</mo>\n <mi>X</mi>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation>$\\Phi _3 (X)$</annotation>\n </semantics></math> of rank 3 quadratic equations and analyze the geometric properties of the irreducible components of <span></span><math>\n <semantics>\n <mrow>\n <msub>\n <mi>Φ</mi>\n <mn>3</mn>\n </msub>\n <mrow>\n <mo>(</mo>\n <mi>X</mi>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation>$\\Phi _3 (X)$</annotation>\n </semantics></math> such as their desingularizations. Additionally, we explore the non-singularity and singularity of these irreducible components of <span></span><math>\n <semantics>\n <mrow>\n <msub>\n <mi>Φ</mi>\n <mn>3</mn>\n </msub>\n <mrow>\n <mo>(</mo>\n <mi>X</mi>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation>$\\Phi _3 (X)$</annotation>\n </semantics></math>.</p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"298 9","pages":"3135-3155"},"PeriodicalIF":0.8000,"publicationDate":"2025-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On rank 3 quadratic equations of Veronese embeddings\",\"authors\":\"Euisung Park, Saerom Sim\",\"doi\":\"10.1002/mana.70028\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>This paper studies the geometric structure of the locus <span></span><math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>Φ</mi>\\n <mn>3</mn>\\n </msub>\\n <mrow>\\n <mo>(</mo>\\n <mi>X</mi>\\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation>$\\\\Phi _3 (X)$</annotation>\\n </semantics></math> of rank 3 quadratic equations of the Veronese variety <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>X</mi>\\n <mo>=</mo>\\n <msub>\\n <mi>ν</mi>\\n <mi>d</mi>\\n </msub>\\n <mrow>\\n <mo>(</mo>\\n <msup>\\n <mi>P</mi>\\n <mi>n</mi>\\n </msup>\\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation>$X = \\\\nu _d ({\\\\mathbb {P}}^n)$</annotation>\\n </semantics></math>. Specifically, we investigate the minimal irreducible decomposition of <span></span><math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>Φ</mi>\\n <mn>3</mn>\\n </msub>\\n <mrow>\\n <mo>(</mo>\\n <mi>X</mi>\\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation>$\\\\Phi _3 (X)$</annotation>\\n </semantics></math> of rank 3 quadratic equations and analyze the geometric properties of the irreducible components of <span></span><math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>Φ</mi>\\n <mn>3</mn>\\n </msub>\\n <mrow>\\n <mo>(</mo>\\n <mi>X</mi>\\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation>$\\\\Phi _3 (X)$</annotation>\\n </semantics></math> such as their desingularizations. Additionally, we explore the non-singularity and singularity of these irreducible components of <span></span><math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>Φ</mi>\\n <mn>3</mn>\\n </msub>\\n <mrow>\\n <mo>(</mo>\\n <mi>X</mi>\\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation>$\\\\Phi _3 (X)$</annotation>\\n </semantics></math>.</p>\",\"PeriodicalId\":49853,\"journal\":{\"name\":\"Mathematische Nachrichten\",\"volume\":\"298 9\",\"pages\":\"3135-3155\"},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2025-08-04\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Mathematische Nachrichten\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1002/mana.70028\",\"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.70028","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
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