Specialization of Mordell-Weil ranks of abelian schemes over surfaces to curves

Using the Shioda-Tate theorem and an adaptation of Silverman's specialization theorem, we reduce the specialization of Mordell-Weil ranks for abelian varieties over fields finitely generated over infinite finitely generated fields k to the specialization theorem for Neron- Severi ranks recently proved by Ambrosi in positive characteristic. More precisely, we prove that after a blow-up of the base surface S, for all vertical curves S-x of a fibration S ? U ? P-k(1) with x from the complement of a sparse subset of |U|, the Mordell-Weil rank of an abelian scheme over S stays the same when restricted to (Sx).