On Vojta's harder implication, height inequalities for admissible pairs, points of bounded degree and Deligne-Mumford stacks
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 (D0,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.
Related
- Existence of coarse moduli superspaces for Deligne--Mumford superstacks
- A Quadratically Enriched Pushforward for Deligne--Mumford Stacks
- Torelli Theorem for moduli stacks of vector bundles on tame Deligne Mumford curves
- Compactification Independence of the Irregular Hodge Filtration on Deligne-Mumford Stacks
- Irregular Hodge Bundles for Deligne--Mumford Landau--Ginzburg Families with Fixed Pole Orders
BibTeX
@misc{grieve2026vojtas,
title = {{On Vojta's harder implication, height inequalities for admissible pairs, points of bounded degree and Deligne-Mumford stacks}},
author = {Nathan Grieve},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.36300},
eprint = {2609.36300},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.36300},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On Vojta's harder implication, height inequalities for admissible pairs, points of bounded degree and Deligne-Mumford stacks
AU - Nathan Grieve
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.36300
UR - https://arxiv.org/abs/2609.36300
AB - 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.
ER -