Elliptic 3-folds in P^2-bundles over surfaces

n this talk we want to study the elliptic fibrations whose total space is a Calabi-Yau variety embedded as an anticanonical divisor in a suitable class of P^2-bundles over a surface. Once we fix the base surface, we will show that the number of such elliptic 3-folds is finite, using only intersection theoretic properties of the base. When the base surface is P^2, we will show that some of these fibrations admit in a very natural way some non-Kodaira fibres.