{"title":"满足一定相切条件的简单阿贝尔变换映射的有限性","authors":"Finn Bartsch","doi":"10.1112/blms.70119","DOIUrl":null,"url":null,"abstract":"<p>We show that given a simple abelian variety <span></span><math>\n <semantics>\n <mi>A</mi>\n <annotation>$A$</annotation>\n </semantics></math> and a normal variety <span></span><math>\n <semantics>\n <mi>V</mi>\n <annotation>$V$</annotation>\n </semantics></math> defined over a finitely generated field <span></span><math>\n <semantics>\n <mi>K</mi>\n <annotation>$K$</annotation>\n </semantics></math> of characteristic zero, the set of non-constant morphisms <span></span><math>\n <semantics>\n <mrow>\n <mi>V</mi>\n <mo>→</mo>\n <mi>A</mi>\n </mrow>\n <annotation>$V \\rightarrow A$</annotation>\n </semantics></math> satisfying certain tangency conditions imposed by a Campana orbifold divisor <span></span><math>\n <semantics>\n <mi>Δ</mi>\n <annotation>$\\Delta$</annotation>\n </semantics></math> on <span></span><math>\n <semantics>\n <mi>A</mi>\n <annotation>$A$</annotation>\n </semantics></math> is finite. To do so, we study the geometry of the scheme <span></span><math>\n <semantics>\n <mrow>\n <msup>\n <munder>\n <mo>Hom</mo>\n <mo>̲</mo>\n </munder>\n <mo>nc</mo>\n </msup>\n <mrow>\n <mo>(</mo>\n <mi>C</mi>\n <mo>,</mo>\n <mrow>\n <mo>(</mo>\n <mi>A</mi>\n <mo>,</mo>\n <mi>Δ</mi>\n <mo>)</mo>\n </mrow>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation>$\\underline{\\operatorname{Hom}}^{\\operatorname{nc}}(C, (A, \\Delta))$</annotation>\n </semantics></math> parameterizing such morphisms from a smooth curve <span></span><math>\n <semantics>\n <mi>C</mi>\n <annotation>$C$</annotation>\n </semantics></math> and show that it admits a quasi-finite non-dominant morphism to <span></span><math>\n <semantics>\n <mi>A</mi>\n <annotation>$A$</annotation>\n </semantics></math>.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 9","pages":"2723-2730"},"PeriodicalIF":0.9000,"publicationDate":"2025-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://londmathsoc.onlinelibrary.wiley.com/doi/epdf/10.1112/blms.70119","citationCount":"0","resultStr":"{\"title\":\"On the finiteness of maps into simple abelian varieties satisfying certain tangency conditions\",\"authors\":\"Finn Bartsch\",\"doi\":\"10.1112/blms.70119\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We show that given a simple abelian variety <span></span><math>\\n <semantics>\\n <mi>A</mi>\\n <annotation>$A$</annotation>\\n </semantics></math> and a normal variety <span></span><math>\\n <semantics>\\n <mi>V</mi>\\n <annotation>$V$</annotation>\\n </semantics></math> defined over a finitely generated field <span></span><math>\\n <semantics>\\n <mi>K</mi>\\n <annotation>$K$</annotation>\\n </semantics></math> of characteristic zero, the set of non-constant morphisms <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>V</mi>\\n <mo>→</mo>\\n <mi>A</mi>\\n </mrow>\\n <annotation>$V \\\\rightarrow A$</annotation>\\n </semantics></math> satisfying certain tangency conditions imposed by a Campana orbifold divisor <span></span><math>\\n <semantics>\\n <mi>Δ</mi>\\n <annotation>$\\\\Delta$</annotation>\\n </semantics></math> on <span></span><math>\\n <semantics>\\n <mi>A</mi>\\n <annotation>$A$</annotation>\\n </semantics></math> is finite. To do so, we study the geometry of the scheme <span></span><math>\\n <semantics>\\n <mrow>\\n <msup>\\n <munder>\\n <mo>Hom</mo>\\n <mo>̲</mo>\\n </munder>\\n <mo>nc</mo>\\n </msup>\\n <mrow>\\n <mo>(</mo>\\n <mi>C</mi>\\n <mo>,</mo>\\n <mrow>\\n <mo>(</mo>\\n <mi>A</mi>\\n <mo>,</mo>\\n <mi>Δ</mi>\\n <mo>)</mo>\\n </mrow>\\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation>$\\\\underline{\\\\operatorname{Hom}}^{\\\\operatorname{nc}}(C, (A, \\\\Delta))$</annotation>\\n </semantics></math> parameterizing such morphisms from a smooth curve <span></span><math>\\n <semantics>\\n <mi>C</mi>\\n <annotation>$C$</annotation>\\n </semantics></math> and show that it admits a quasi-finite non-dominant morphism to <span></span><math>\\n <semantics>\\n <mi>A</mi>\\n <annotation>$A$</annotation>\\n </semantics></math>.</p>\",\"PeriodicalId\":55298,\"journal\":{\"name\":\"Bulletin of the London Mathematical Society\",\"volume\":\"57 9\",\"pages\":\"2723-2730\"},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2025-06-13\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://londmathsoc.onlinelibrary.wiley.com/doi/epdf/10.1112/blms.70119\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Bulletin of the London Mathematical Society\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/blms.70119\",\"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":"Bulletin of the London Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/blms.70119","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
On the finiteness of maps into simple abelian varieties satisfying certain tangency conditions
We show that given a simple abelian variety and a normal variety defined over a finitely generated field of characteristic zero, the set of non-constant morphisms satisfying certain tangency conditions imposed by a Campana orbifold divisor on is finite. To do so, we study the geometry of the scheme parameterizing such morphisms from a smooth curve and show that it admits a quasi-finite non-dominant morphism to .
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