关于Vojta的更难蕴含、可容许对的高度不等式、有界次数点与Deligne-Mumford栈
On Vojta's harder implication, height inequalities for admissible pairs, points of bounded degree and Deligne-Mumford stacks
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中文总结 AI 辅助
本研究扩展可容许对概念，证明Vojta的更难蕴含，并针对一般型Deligne-Mumford栈推导出有界次数整点的Bombieri-Lang猜想形式，结合切片定理与Silverman不等式。
中文摘要 AI 辅助
我们扩展了来自文献[Levin:GCD]的\emph{可容许对}概念，并探讨了它与Vojta在文献[Vojta:1998]中预测的主要丢番图算术不等式（含判别式项及针对有界次数点的形式）之间的关系。在此背景下，除其他新结果外，我们证明了一个\emph{更难蕴含}，这符合Vojta处理abc猜想的方法精神（源自文献[Vojta:1998]）。作为我们的主要结果及我们观点的应用，我们针对具有射影粗模空间的某些一般型非奇异Deligne-Mumford栈，推导出了关于有界次数的$(D_0,S)$-整点的Bombieri-Lang猜想的一种形式。一个关键输入是Abramovich和Várilly-Alvarado在文献[Abramovich:VarillyAlvarado:Pera:2017]中获得的Deligne-Mumford栈切片定理，该定理建立在Kresch和Vistoli的早期工作[Kresch:Vistoli:2004]之上。另一个重要成分是Silverman在文献[Silverman:1984]中的不等式，该不等式用射影空间中点的高度来界定其判别式。作为我们结果的例证，我们在Abramovich和Harris的引人注目的工作[Abramovich:Harris:1991]及其他工作的背景下讨论了它们。
英文摘要
We expand on the concept of \emph{admissible pairs} from \cite{Levin:GCD} and explore its relation with the main Diophantine arithmetic inequalities, with discriminant term and for points of bounded degree, that have been predicted by Vojta \cite{Vojta:1998}. In this context, among other new results, we prove a \emph{harder implication} which is in the spirit of Vojta's approach to the abc Conjecture (from \cite{Vojta:1998}). As our main result, and application of our viewpoint here, we deduce for the case of certain general type nonsingular Deligne-Mumford stacks, with projective course moduli space, a form of the Bombieri-Lang Conjecture for $(D_0,S)$-integral points of bounded degree. A key input for this is a slicing theorem, for Deligne-Mumford stacks, that was obtained by Abramovich and Várilly-Alvarado, \cite{Abramovich:VarillyAlvarado:Pera:2017}, and building on earlier work of Kresch and Vistoli \cite{Kresch:Vistoli:2004}. Another important ingredient is an inequality of Silverman, from \cite{Silverman:1984}, which bounds the discriminant of points in projective space in terms of their heights. As an illustration of our results, we discuss them within the context of the interesting work of Abramovich and Harris \cite{Abramovich:Harris:1991} and others.
发表机构
- National Taiwan University（台湾大学）
- Carleton University（卡尔顿大学）
- Université du Québec à Montréal（蒙特利尔大学）
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