Introduction

In this chapter we study heights from a geometric point of view.

We begin with the important Section 2.2 introducing local Weil heights associated to Cartier divisors on a projective variety X, and studying their properties. These considerations are given here only for projective varieties, where the treatment is simpler.

Section 2.3 studies global Weil heights and their equivalence classes up to bounded functions.

In Section 2.4, we study the height on a projective variety induced by the height in the ambient projective space and in particular we prove the important Northcott's theorem on the finiteness of the number of points of bounded degree and bounded height in a fixed projective space.

These three sections are very important for the handling of heights in diophantine geometry and are required from Chapter 9 onwards.

In Section 2.5, which contains new material, the notion of presentation of a projective variety is introduced and explicit comparison theorems for the heights of a variety X in two different projective embeddings are given, in terms of presentations of these embeddings. This section may be skipped in a first reading. It will be used only partially in Section 11.7 and implicitly in questions dealing with effectivity.

Sections 2.6 and 2.7 extend the results obtained on local and global Weil heights to the associated heights of locally bounded metrized line bundles on a complete variety. They will be also used in the second half of the book.