Threefolds whose hyperplane sections are elliptic surfaces. (English) Zbl 0626.14031

D’Souza, Harry

Pac. J. Math. 134, No. 1, 57-78 (1988).

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 \(\kappa (S)=1\).We use the logarithmic Kodaira dimension of \((X,S)\) as the basis of classification. Sommese studied such pairs, and he has shown that such pairs \((X,S)\) can be reduced to \((X',S')\), where \(S'\) is ample in \(X'\), and \(S'\) is a minimal model

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of \(S\). In the case when \(S\) is elliptic, with \(h^{1,0}(S)\neq 0\) he showed that one obtains a surjective morphism p, from X onto a smooth curve

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\(Y\) 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 \(h^{1,0}(S)\). We state these results in section 0.
We show that the general fibres of p are del Pezzo surfaces

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and classify these in the case where they are of degrees 1, 2, 3, 4, 7, 8 and 9. We also show in the degree 9 case, it is indeed a \({\mathbb{P}}^ 2\)-bundle over Y. In the degree 8 (\(\cong {\mathbb{P}}^ 1\times {\mathbb{P}}^ 1)\) case we have a birational morphism to a \({\mathbb{P}}^ 2\)-bundle.

MSC:

14J30	\(3\)-folds
14C20	Divisors, linear systems, invertible sheaves
14E05	Rational and birational maps

Keywords:

threefold; smooth ample divisor; logarithmic Kodaira dimension; elliptic fibration