On Vojta's harder implication, height inequalities for admissible pairs, points of bounded degree and Deligne-Mumford stacks

Source abstract

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.