The height of an algebraic point is an important tool in Diophantine geometry. If $\overline{L}$ is a line bundle with an adelic metric on $X$, then for any point $P\in X(F)$, viewed as a closed subscheme of $X$, one has

$$h_{\overline{L}}(P)=({\widehat{c}}_{1}(\overline{L})|P)=\sum_{v}\log\left\|{s}\right\|^{-1}_{v}(P),$$

where $s$ is any meromorphic section on $L$ which has neither a zero nor a pole at $P$. More generally, let $P\in X(\overline{F})$ be an algebraic point and let $[P]$ be the corresponding closed point of $X$. Then,

$$h_{\overline{L}}(P)=\frac{1}{[F(P):F]}({\widehat{c}}_{1}(\overline{L})|[P])$$

is the height of $P$ with respect to the metrized line bundle $\overline{L}$. In fact, restricted to points, these definitions apply to any, not necessary admissible,

Observe also the following functorial property of the height : If $f\colon Y\rightarrow X$ is a morphism and $P\in Y(\overline{F})$, then $h_{f^{*}\overline{L}}(P)=h_{\overline{L}}(f(P))$. Finally, recall that if $F$ is a global field, then the height with respect to a metrized ample line bundle $\overline{L}$ satisfies Northcott’s finiteness property : for any integers $d$ and $B$, there are only finitely many points $P\in X(\overline{F})$ such that $[F(P):F]\leq d$ and $h_{\overline{L}}(P)\leq B$.