2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/150232It is proved that on a smooth algebraic variety, fibered into cubic surfaces over the projective line and sufficiently ``twisted'' over the base, there is only one pencil of rational surfaces -- that is, this very pencil of cubics. In particular, this variety is non-rational; moreover, it can not be fibered into rational curves. The proof is obtained by means of the method of maximal singularities.29 pages, latex, to appear in IzvestiaAlgebraic Geometrytext