On the heights of algebraic points on curves over number fields

Let X be a semi-stable regular curve over the spectrum S of the integers in a number field F , and L̄ = (L, h) an hermitian line bundle on X , i.e. L is an algebraic line bundle on X and h is a smooth hermitian metric (invariant by complex conjugation) on the restriction of L to the set X(C) of complex points of X . In this paper we are interested in the height hL̄(D) of irreducible divisors D on X which are flat over S, i.e. the arithmetic degree of the restriction of L̄ to D. First we assume that the degree deg(L) of L on the generic fiber XF is positive and we denote by L̄ · L̄ ∈ R the self-intersection of the first arithmetic Chern class of L̄. Define