2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/165263We prove that for any of a wide class of elliptic surfaces $X$ defined over a number field $k$, if there is an algebraic point on $X$ that lies on only finitely many rational curves, then there is an algebraic point on $X$ that lies on no rational curves. In particular, our theorem applies to a large class of elliptic $K3$ surfaces, which relates to a question posed by Bogomolov in 1981. We apply our results to construct an explicit algebraic point on a $K3$ surface that does not lie on any smooth rational curves.10 pages, no figures. An explicit construction of an algebraic point lying on no smooth rational curves has been added to the end, and there have been minor revisions to the rest of the paperAlgebraic GeometryNumber Theory14G05text