{"title":"正特征中的 geproci 性质","authors":"Jake Kettinger","doi":"10.1090/proc/16809","DOIUrl":null,"url":null,"abstract":"<p>The geproci property is a recent development in the world of geometry. We call a set of points <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper Z subset-of-or-equal-to double-struck upper P Subscript k Superscript 3\"> <mml:semantics> <mml:mrow> <mml:mi>Z</mml:mi> <mml:mo>⊆</mml:mo> <mml:msubsup> <mml:mrow> <mml:mi mathvariant=\"double-struck\">P</mml:mi> </mml:mrow> <mml:mi>k</mml:mi> <mml:mn>3</mml:mn> </mml:msubsup> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">Z\\subseteq \\mathbb {P}_k^3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> an <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis a comma b right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>a</mml:mi> <mml:mo>,</mml:mo> <mml:mi>b</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">(a,b)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-geproci set (for GEneral PROjection is a Complete Intersection) if its projection from a general point <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper P\"> <mml:semantics> <mml:mi>P</mml:mi> <mml:annotation encoding=\"application/x-tex\">P</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to a plane is a complete intersection of curves of degrees <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"a less-than-or-equal-to b\"> <mml:semantics> <mml:mrow> <mml:mi>a</mml:mi> <mml:mo>≤</mml:mo> <mml:mi>b</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">a\\leq b</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Nondegenerate examples known as grids have been known since 2011. Nondegenerate nongrids were first described in 2018, working in characteristic 0. Almost all of these new examples are of a special kind called half grids.</p> <p>In this paper, based partly on the author’s thesis, we use a feature of geometry in positive characteristic to give new methods of producing geproci half grids and non-half grids.</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-02-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"The geproci property in positive characteristic\",\"authors\":\"Jake Kettinger\",\"doi\":\"10.1090/proc/16809\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>The geproci property is a recent development in the world of geometry. We call a set of points <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper Z subset-of-or-equal-to double-struck upper P Subscript k Superscript 3\\\"> <mml:semantics> <mml:mrow> <mml:mi>Z</mml:mi> <mml:mo>⊆</mml:mo> <mml:msubsup> <mml:mrow> <mml:mi mathvariant=\\\"double-struck\\\">P</mml:mi> </mml:mrow> <mml:mi>k</mml:mi> <mml:mn>3</mml:mn> </mml:msubsup> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">Z\\\\subseteq \\\\mathbb {P}_k^3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> an <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"left-parenthesis a comma b right-parenthesis\\\"> <mml:semantics> <mml:mrow> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:mi>a</mml:mi> <mml:mo>,</mml:mo> <mml:mi>b</mml:mi> <mml:mo stretchy=\\\"false\\\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">(a,b)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-geproci set (for GEneral PROjection is a Complete Intersection) if its projection from a general point <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper P\\\"> <mml:semantics> <mml:mi>P</mml:mi> <mml:annotation encoding=\\\"application/x-tex\\\">P</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to a plane is a complete intersection of curves of degrees <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"a less-than-or-equal-to b\\\"> <mml:semantics> <mml:mrow> <mml:mi>a</mml:mi> <mml:mo>≤</mml:mo> <mml:mi>b</mml:mi> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">a\\\\leq b</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Nondegenerate examples known as grids have been known since 2011. Nondegenerate nongrids were first described in 2018, working in characteristic 0. Almost all of these new examples are of a special kind called half grids.</p> <p>In this paper, based partly on the author’s thesis, we use a feature of geometry in positive characteristic to give new methods of producing geproci half grids and non-half grids.</p>\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2024-02-29\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1090/proc/16809\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/proc/16809","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
摘要
geproci 属性是几何学领域的最新发展。如果一个点集 Z ⊆ P k 3 Z\subseteq \mathbb {P}_k^3 是一个（a , b ）（a,b）-geproci 集（GEneral PROjection is a Complete Intersection 的缩写），而它从一般点 P P 到平面的投影是 a≤b a\leq b 的度数的曲线的完全交集，我们就称这个点集为 geproci 集。早在 2011 年，人们就知道了被称为网格的非enerate 例子。2018 年首次描述了非enerate 非网格，在特征 0 下工作。几乎所有这些新例子都属于一种特殊类型，称为半网格。在本文中，我们部分基于作者的论文，利用正特征几何的一个特点，给出了产生geproci半网格和非半网格的新方法。
The geproci property is a recent development in the world of geometry. We call a set of points Z⊆Pk3Z\subseteq \mathbb {P}_k^3 an (a,b)(a,b)-geproci set (for GEneral PROjection is a Complete Intersection) if its projection from a general point PP to a plane is a complete intersection of curves of degrees a≤ba\leq b. Nondegenerate examples known as grids have been known since 2011. Nondegenerate nongrids were first described in 2018, working in characteristic 0. Almost all of these new examples are of a special kind called half grids.
In this paper, based partly on the author’s thesis, we use a feature of geometry in positive characteristic to give new methods of producing geproci half grids and non-half grids.
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