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.