Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2021
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2108.04411 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
Table of Contents:
- In this paper we investigate a family of algebraic stacks, the so-called stacky curves, in the context of the general theory of heights on algebraic stacks due to Ellenberg, Satriano, and Zureick-Brown. We first give an elementary construction of a height which is seen to be dual to theirs. Next we count rational points having bounded E-S-ZB height on a particular stacky curve, answering a question of Ellenberg, Satriano, and Zureick-Brown. We then show that when the Euler characteristic of stacky curves is non-positive, that the E-S-ZB height coming from the anti-canonical divisor class fails to have the Northcott property. Next we prove a generalized version of a conjecture of Vojta, applied to stacky curves with negative Euler characteristic and coarse space $\mathbb{P}^1$, is equivalent to the $abc$-conjecture. Finally, we prove that in the negative characteristic case the purely "stacky" part of the E-S-ZB height exhibits the Northcott property.