  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.
