2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/77909We study the geometry and cohomology of semiample hypersurfaces in toric varieties. Such hypersurfaces generalize the MPCP-desingularizations of Calabi-Yau ample hypersurfaces in the Batyrev mirror construction. We study the topological cup product on the middle cohomology of semiample hypersurfaces. In particular, we obtain a complete algebraic description of the middle cohomology of regular semiample hypersurfaces in 4-dimensional simplicial toric varieties what would be interesting for physics.Some corrections made (Lemma 5.7, Theorem 5.9)Algebraic Geometry14M25text