J. Naranjo, –. Ortega
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{"title":"全球普瑞姆-托瑞利的双重覆盖至少延伸到六个点","authors":"J. Naranjo, –. Ortega","doi":"10.1090/jag/779","DOIUrl":null,"url":null,"abstract":"<p>We prove that the ramified Prym map <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper P Subscript g comma r\">\n <mml:semantics>\n <mml:msub>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">P</mml:mi>\n </mml:mrow>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi>g</mml:mi>\n <mml:mo>,</mml:mo>\n <mml:mi>r</mml:mi>\n </mml:mrow>\n </mml:msub>\n <mml:annotation encoding=\"application/x-tex\">\\mathcal P_{g, r}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> which sends a covering <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"pi colon upper D long right-arrow upper C\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>π<!-- π --></mml:mi>\n <mml:mo>:</mml:mo>\n <mml:mi>D</mml:mi>\n <mml:mo stretchy=\"false\">⟶<!-- ⟶ --></mml:mo>\n <mml:mi>C</mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\pi :D\\longrightarrow C</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> ramified in <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"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> points to the Prym variety <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper P left-parenthesis pi right-parenthesis colon-equal upper K e r left-parenthesis upper N m Subscript pi Baseline right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>P</mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>π<!-- π --></mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mo>≔</mml:mo>\n <mml:mi>K</mml:mi>\n <mml:mi>e</mml:mi>\n <mml:mi>r</mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>N</mml:mi>\n <mml:msub>\n <mml:mi>m</mml:mi>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi>π<!-- π --></mml:mi>\n </mml:mrow>\n </mml:msub>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">P(\\pi )≔Ker(Nm_{\\pi })</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> is an embedding for all <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"r greater-than-or-equal-to 6\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>r</mml:mi>\n <mml:mo>≥<!-- ≥ --></mml:mo>\n <mml:mn>6</mml:mn>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">r\\ge 6</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> and for all <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"g left-parenthesis upper C right-parenthesis greater-than 0\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>g</mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>C</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mo>></mml:mo>\n <mml:mn>0</mml:mn>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">g(C)>0</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. Moreover, by studying the restriction to the locus of coverings of hyperelliptic curves, we show that <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper P Subscript g comma 2\">\n <mml:semantics>\n <mml:msub>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">P</mml:mi>\n </mml:mrow>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi>g</mml:mi>\n <mml:mo>,</mml:mo>\n <mml:mn>2</mml:mn>\n </mml:mrow>\n </mml:msub>\n <mml:annotation encoding=\"application/x-tex\">\\mathcal P_{g, 2}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> and <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper P Subscript g comma 4\">\n <mml:semantics>\n <mml:msub>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">P</mml:mi>\n </mml:mrow>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi>g</mml:mi>\n <mml:mo>,</mml:mo>\n <mml:mn>4</mml:mn>\n </mml:mrow>\n </mml:msub>\n <mml:annotation encoding=\"application/x-tex\">\\mathcal P_{g, 4}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> have positive dimensional fibers.</p>","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2020-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"11","resultStr":"{\"title\":\"Global Prym-Torelli for double coverings ramified in at least six points\",\"authors\":\"J. Naranjo, –. Ortega\",\"doi\":\"10.1090/jag/779\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We prove that the ramified Prym map <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"script upper P Subscript g comma r\\\">\\n <mml:semantics>\\n <mml:msub>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi class=\\\"MJX-tex-caligraphic\\\" mathvariant=\\\"script\\\">P</mml:mi>\\n </mml:mrow>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi>g</mml:mi>\\n <mml:mo>,</mml:mo>\\n <mml:mi>r</mml:mi>\\n </mml:mrow>\\n </mml:msub>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\mathcal P_{g, r}</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> which sends a covering <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"pi colon upper D long right-arrow upper C\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi>π<!-- π --></mml:mi>\\n <mml:mo>:</mml:mo>\\n <mml:mi>D</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">⟶<!-- ⟶ --></mml:mo>\\n <mml:mi>C</mml:mi>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\pi :D\\\\longrightarrow C</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> ramified in <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"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> points to the Prym variety <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper P left-parenthesis pi right-parenthesis colon-equal upper K e r left-parenthesis upper N m Subscript pi Baseline right-parenthesis\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi>P</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mi>π<!-- π --></mml:mi>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n <mml:mo>≔</mml:mo>\\n <mml:mi>K</mml:mi>\\n <mml:mi>e</mml:mi>\\n <mml:mi>r</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mi>N</mml:mi>\\n <mml:msub>\\n <mml:mi>m</mml:mi>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi>π<!-- π --></mml:mi>\\n </mml:mrow>\\n </mml:msub>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">P(\\\\pi )≔Ker(Nm_{\\\\pi })</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> is an embedding for all <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"r greater-than-or-equal-to 6\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi>r</mml:mi>\\n <mml:mo>≥<!-- ≥ --></mml:mo>\\n <mml:mn>6</mml:mn>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">r\\\\ge 6</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> and for all <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"g left-parenthesis upper C right-parenthesis greater-than 0\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi>g</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mi>C</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n <mml:mo>></mml:mo>\\n <mml:mn>0</mml:mn>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">g(C)>0</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>. Moreover, by studying the restriction to the locus of coverings of hyperelliptic curves, we show that <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"script upper P Subscript g comma 2\\\">\\n <mml:semantics>\\n <mml:msub>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi class=\\\"MJX-tex-caligraphic\\\" mathvariant=\\\"script\\\">P</mml:mi>\\n </mml:mrow>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi>g</mml:mi>\\n <mml:mo>,</mml:mo>\\n <mml:mn>2</mml:mn>\\n </mml:mrow>\\n </mml:msub>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\mathcal P_{g, 2}</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> and <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"script upper P Subscript g comma 4\\\">\\n <mml:semantics>\\n <mml:msub>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi class=\\\"MJX-tex-caligraphic\\\" mathvariant=\\\"script\\\">P</mml:mi>\\n </mml:mrow>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi>g</mml:mi>\\n <mml:mo>,</mml:mo>\\n <mml:mn>4</mml:mn>\\n </mml:mrow>\\n </mml:msub>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\mathcal P_{g, 4}</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> have positive dimensional fibers.</p>\",\"PeriodicalId\":54887,\"journal\":{\"name\":\"Journal of Algebraic Geometry\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2020-05-22\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"11\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Algebraic Geometry\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1090/jag/779\",\"RegionNum\":1,\"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":"Journal of Algebraic Geometry","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/jag/779","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 11
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