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 Mordell–Weil ranks for abelian varieties over fields finitely generated infinite [Formula: see text] to Néron–Severi recently proved by Ambrosi in positive characteristic. More precisely, prove that after a blow-up base surface text], all vertical curves fibration with from complement sparse subset rank scheme stays same when restricted text].