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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 dimension 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 (X′,S′), where
S′ is ample in X′, and S′ is minimal model of S. In the case when S is
elliptic, with h1,0(S)≠0 he showed that one obtains a surjective morphism p,
from X onto a smooth curve Y such that this morphism restricted to S is a
reduced elliptic fibration. Shepherd-Barron proved the same result using
Mori’s methods without the restriction on h1,0(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 over Y . In the degree 8 (≅P1 ×P1) case we have a
birational morphism to a P2-bundle.
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