ICM-90 Satellite Conference Proceedings pp 126-137 | Cite as

# Elliptic Fibrations over Surfaces I

## Abstract

The structure of a degeneration of elliptic curves is completely clarified by the work of Kodaira [7] and this is very important in the classification theory of compact complex surfaces. We are now interested in the classification of threefolds. In this paper, we shall study proper surjective morphisms over surfaces: *f*: *X* → *S* whose general fibers are elliptic curves. These *f* are called elliptic fibrations over surfaces *S*. We want to treat in general situation, but here we assume that *S* is a nonsingular surface, *f* is smooth outside of a normal crossing divisor of *S* and that *f* is a locally projective morphism. If *f*: *X* → *S* does not satisfy this assumption, after suitable blow-up of *S* and *X*, the result fibration satisfies the assumption. Note that *f* may not be a flat morphism. Further we restrict ourselves to the study of local structure of the fibration. Namely we assume that the base surface *S* is the two-dimensional unit polydisc Δ^{2}:= {(t_{1}, t_{2}) ∈ C^{2}||t_{1}| < 1, |t_{2}| < 1} or more precisely, the germ (Δ^{2},0). Here 0 denotes the origin (0,0) ∈ Δ^{2}. And also we assume that *f* is a projective morphism and is smooth outside of {t_{1} = 0} U {t_{2} = 0}.

## Keywords

Elliptic Curf Singular Locus Elliptic Surface Crepant Resolution Central Fiber## Preview

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