What is the "geometric height" mentioned by Moriwaki?

Question

Let $K$ be a finitely generated field over $\mathbb{Q}$ of transcendence degree 1, and take a curve $C$ over a number field $k$ such that $k(C)=K$. In "Arithmetic height functions over finitely generated fields" (Invent. Math. 140 (2000), no. 1, 101–142), Moriwaki states that one can use points on $C$ to define non-archimedean valuations that give a "geometric height" $h:\mathbb{P}^n(\overline{K})\to\mathbb{R}$. Moriwaki then writes:

It is well known that this height function can be given in terms of the usual intersection theory, so that it is rather easy to handle it.

Question 1: What is the intersection theoretic formulation of this geometric height? I think it should be given by taking the degree of some line bundle, since Moriwaki later defines a general height function in terms of the arithmetic/Arakelov degree of various Hermitian line bundles.

I've tried searching for other papers and books that discuss "geometric heights." They seem to be referenced fairly frequently, but I could not find any references where they are explicitly defined.

Question 2: More generally, is there a "classical" geometric height $h:X(\overline{K})\to\mathbb{R}$ for $X$ a projective variety? I guess if the line bundles arising in Question 1 come from maps $C\to\mathbb{P}^n$, then the answer may be no.

Answer 1

The geometric height is easiest to define for points on $\mathbb P^n(K)$. These define maps $C \to \mathbb P^n$ and we take the line bundle $\mathcal O(1)$ on $\mathbb P^n$, pull back to $C$, and take the degree.

We can express this with valuations by fixing coordinates $(a_0,\dots, a_n)$ and taking $-\sum_v \min ( v(a_0),\dots, v(a_n) ) $ where the $v$ are valuations of $K$ trivial on $k$.

General points of $\mathbb P^n(\overline{K})$ are defined over $k(C')$ for some cover $C'$ of $C$, say of degree $d$. We take $\mathcal O(1)$ on $\mathbb P^n$, pull back to $C'$, take its degree, and then probably divide by $C$ to normalize.

Alternatively, we can view the graph of a point of $\mathbb P^n(\overline{K})$ as a curve in $C \times \mathbb P^n$ and take the intersection product with a hyperplane class of $\mathbb P^n$ (then normalize by dividing by the intersection product with a degree $1$ divisor from $C$).

The same works for an arbitrary projective variety $X$ as long as we fix an ample line bundle on $X$, but will depend on the choice of the line bundle.