2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/253We construct a (smooth, projective) surface over the field of rational numbers, which is a counterexample to the Hasse principle not accounted for by the Manin obstruction. The construction relies on the classical 4-descent on elliptic curves. By combining the Manin obstruction with descent we define a refinement of the Manin obstruction which for surfaces of the type considered suffices to explain the absence of a rational point.Plain TeX, 15 pages, an appendix by S. SiksekAlgebraic Geometrytext