On the number of points of bounded height on arithmetic projective spaces

Let K be a number field and O(K) its ring of integers. Let (E) over bar be a Hermitian vector bundle over Spec(O(K)). In the first part of this paper we estimate the number of points of bounded height in P((E) over bar)(K) (generalizing a result by Schanuel). We give then some applications: we estimate the number of hyperplanes and hypersurfaces of degree d > 1 in P((E) over bar) of bounded height and containing a fixed linear subvariety and we estimate the number of points of height, with respect to the anticanonical line bundle, less then T (when T goes to infinity) of P(K)(N) blown up at a linear subspace of codimension two.