Let $X$ be a smooth, projective, geometrically irreducible curve defined over a number field, and let $D$ a divisor on $X$. To these data, we can associate a height function on the $\Bbb{\overline{Q}}$-rational points of $X$ $$ \operatorname{ht}_D: X(\Bbb{\overline{Q}}) \rightarrow \Bbb{R} $$ as follows. Write $D=D_1- D_2$, with $D_1,D_2$ very ample divisors on $X$, and let $\phi_{D_1},\phi_{D_2}: X \hookrightarrow \Bbb{P}^n$ be embeddings determined by $D_1,D_2$. Then $$ \operatorname{ht}_D(P):=\operatorname{ht}_{D_1}(P)-\operatorname{ht}_{D_2}(P):=\operatorname{h}(\phi_{D_1}(P))-\operatorname{h}(\phi_{D_2}(P)) $$ where $\operatorname{h}: \Bbb{P}^n \rightarrow \Bbb{R}$ is the usual height function on projective space. One can check that, up to a bounded function, $\operatorname{ht}_D$ does not depend on the choices of $D_1,D_2$.
Now I know that if $\deg(D) >0$, the height function $\operatorname{ht}_D$ is bounded from below, and if $\deg(D) < 0$, then $\operatorname{ht}_D$ is bounded from above.
$\textbf{Question:}$ What happens if $\deg(D)=0$? Is there a curve $X$ as above and a divisor $D$, with $\deg(D)=0$ such that $\operatorname{ht}_D$ is not bounded from below/above?
The question as to whether $\operatorname{ht}_D$ for a degree zero divisor $D$ is bounded depends on the class of $D$ in the Picard group of $X$.
More generally, the theorem is true for an arbitrary smooth, projective variety.
The "if" part is clear, and the only if part is proved in
The most important fact for the proof seems to be that the Jacobian of $X$ is principally polarised, i.e. it is isomorphic to its dual abelian variety.
The above theorem is somewhat interesting in my opinion, partly because it resembles the well-known result that on an abelian variety $A$ a point is a torsion point, if and only if its Néron-Tate height (associated to an ample, symmetric divisor) vanishes.