# What is the “geometric height” mentioned by Moriwaki?

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.

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.