Threefolds whose hyperplane sections are elliptic surfaces

In this paper we classify pairs (X, S) where X is a smooth complex projective threefold and S is a smooth ample divisor in X. Moreover S is elliptic and κ(S) = 1. We use the logarithmic Kodaira dimen-sion of (X, S) as the basis of classification. Sommese studied such pairs in "The birational theory of hyperplane sections of projective threefolds " where he showed that such pairs (X, S) can be reduced to (Xf, S')9 where S ' is ample in X \ and S ' is minimal model of 5. In the case when S is elliptic, with hl0(S) Φ 0 he showed that one obtains a surjective morphism /?, from X onto a smooth curve Y such that this morphism restricted to S is a reduced elliptic fibra-tion. Shepherd-Barron proved the same result using Mori's methods without the restriction on hι0(S). We state these results in §0. We show that the general fibres of p are del Pezzo surfaces and classify these in the case where they are of degrees 1, 2, 3, 4, 7, 8 and 9. We show that in the degree 9 case that it is indeed a P2-bundle ove