{"title":"论曲线的𝑝-rank","authors":"Sadik Terzİ","doi":"10.1090/proc/16841","DOIUrl":null,"url":null,"abstract":"<p>In this paper, we are concerned with the computations of the <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p\"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding=\"application/x-tex\">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-rank of curves in two different setups. We first work with complete intersection varieties in <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"bold upper P Superscript n Baseline for n greater-than-or-equal-to 2\"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi mathvariant=\"bold\">P</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:mtext> for </mml:mtext> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">\\mathbf {P}^n \\text { for } n\\ge 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and compute explicitly the action of Frobenius on the top cohomology group. In case of curves and surfaces, this information suffices to determine if the variety is ordinary. Next, we consider curves on more general surfaces with <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p Subscript g Baseline left-parenthesis upper S right-parenthesis equals 0 equals q left-parenthesis upper S right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>p</mml:mi> <mml:mi>g</mml:mi> </mml:msub> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>S</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mo>=</mml:mo> <mml:mi>q</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>S</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">p_g(S) = 0 = q(S)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such as Hirzebruch surfaces and determine <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p\"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding=\"application/x-tex\">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-rank of curves on Hirzebruch surfaces.</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On the 𝑝-rank of curves\",\"authors\":\"Sadik Terzİ\",\"doi\":\"10.1090/proc/16841\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>In this paper, we are concerned with the computations of the <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"p\\\"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding=\\\"application/x-tex\\\">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-rank of curves in two different setups. We first work with complete intersection varieties in <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"bold upper P Superscript n Baseline for n greater-than-or-equal-to 2\\\"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi mathvariant=\\\"bold\\\">P</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:mtext> for </mml:mtext> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">\\\\mathbf {P}^n \\\\text { for } n\\\\ge 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and compute explicitly the action of Frobenius on the top cohomology group. In case of curves and surfaces, this information suffices to determine if the variety is ordinary. Next, we consider curves on more general surfaces with <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"p Subscript g Baseline left-parenthesis upper S right-parenthesis equals 0 equals q left-parenthesis upper S right-parenthesis\\\"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>p</mml:mi> <mml:mi>g</mml:mi> </mml:msub> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:mi>S</mml:mi> <mml:mo stretchy=\\\"false\\\">)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mo>=</mml:mo> <mml:mi>q</mml:mi> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:mi>S</mml:mi> <mml:mo stretchy=\\\"false\\\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">p_g(S) = 0 = q(S)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such as Hirzebruch surfaces and determine <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"p\\\"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding=\\\"application/x-tex\\\">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-rank of curves on Hirzebruch surfaces.</p>\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2024-03-29\",\"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/16841\",\"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/16841","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
在本文中，我们关注两种不同情况下曲线 p p -rank 的计算。我们首先处理 n ≥ 2 \mathbf {P}^n \text { for } n\ge 2 的 P n 中的完全交集品种，并明确计算 Frobenius 对顶同调群的作用。在曲线和曲面的情况下，这些信息足以确定该变化是否普通。接下来，我们考虑更一般的曲面上的曲线，即 p g ( S ) = 0 = q ( S ) p_g(S) = 0 = q(S)，如希尔泽布鲁赫曲面，并确定希尔泽布鲁赫曲面上曲线的 p p -rank。
In this paper, we are concerned with the computations of the pp-rank of curves in two different setups. We first work with complete intersection varieties in Pn for n≥2\mathbf {P}^n \text { for } n\ge 2 and compute explicitly the action of Frobenius on the top cohomology group. In case of curves and surfaces, this information suffices to determine if the variety is ordinary. Next, we consider curves on more general surfaces with pg(S)=0=q(S)p_g(S) = 0 = q(S) such as Hirzebruch surfaces and determine pp-rank of curves on Hirzebruch surfaces.
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