I will report on joint work with Martin Widmer. Let X be a smooth projective variety over a number field K. We prove a Bertini-type theorem with explicit control of the genus, degree, height, and field of definition of the constructed curve on X. As a consequence we provide a general strategy to reduce certain height and rank estimates on abelian varieties over a number field K to the case of Jacobian varieties defined over a suitable extension of K. We will give examples where the strategy works well!