全球普瑞姆-托瑞利的双重覆盖至少延伸到六个点

IF 0.9 1区数学 Q2 MATHEMATICS

Journal of Algebraic Geometry Pub Date : 2020-05-22 DOI:10.1090/jag/779

J. Naranjo, –. Ortega

{"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</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</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</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. 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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

摘要

我们证明了分支Prym映射P g, r \mathcal p＿{G, r} 它发出一个覆盖π: D \pi : d\longrightarrow C在r中的分支r指向P(π)，其中K e r (N m π) P(\pi )对象是Ker(Nm＿{\pi })是对所有r≥6r的嵌入\ge 对于所有g(C)>0 g(C)>0。此外，通过研究超椭圆曲线覆盖轨迹的限制，我们证明了P g, 2 \mathcal p＿{G, 2} P g, 4 \mathcal p＿{G, 4} 有正维纤维。

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Global Prym-Torelli for double coverings ramified in at least six points

We prove that the ramified Prym map P g , r \mathcal P_{g, r} which sends a covering π : D ⟶ C \pi :D\longrightarrow C ramified in r r points to the Prym variety P ( π ) ≔ K e r ( N m π ) P(\pi )≔Ker(Nm_{\pi }) is an embedding for all r ≥ 6 r\ge 6 and for all g ( C ) > 0 g(C)>0 . Moreover, by studying the restriction to the locus of coverings of hyperelliptic curves, we show that P g , 2 \mathcal P_{g, 2} and P g , 4 \mathcal P_{g, 4} have positive dimensional fibers.

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来源期刊

Journal of Algebraic Geometry 数学-数学

CiteScore

2.70

自引率

5.60%

发文量

审稿时长

>12 weeks

期刊介绍： The Journal of Algebraic Geometry is devoted to research articles in algebraic geometry, singularity theory, and related subjects such as number theory, commutative algebra, projective geometry, complex geometry, and geometric topology. This journal, published quarterly with articles electronically published individually before appearing in an issue, is distributed by the American Mathematical Society (AMS). In order to take advantage of some features offered for this journal, users will occasionally be linked to pages on the AMS website.