3
$\begingroup$

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.

$\endgroup$
5
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.