2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/113807The theorem referred to in the title is a technical result that is needed for the classification of elliptic and K3 fibrations birational to Fano 3-fold hypersurfaces in weighted projective space. We present a complete proof of the Curve Exclusion Theorem, which first appeared in the author's unpublished PhD thesis [Ry02] and has since been relied upon in solutions to many cases of the fibration classification problem. We give examples of these solutions and discuss them briefly.26 pagesAlgebraic Geometry14J30; 14E05text