2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/141061These are the substantially expanded notes of the lectures of JK at the summer school "Higher-Dimensional Geometry over Finite Fields" in Göttingen, June 2007. The first part gives an overview of the methods. The main new result is the construction of rational curves passing through a given collection of points on smooth cubic hypersurfaces over finite fields.Algebraic GeometryNumber Theory14G05, 14G15, 11D25, 14M20 (Primary); 14G20, 11G25 (Secondary)text