The Singularities of the parameter surface of a minimal elliptic threefold

Details The Singularities of the parameter surface of a minimal elliptic threefold The Singularities of the parameter surface of a minimal elliptic threefold

1992-02-21

arxiv.org/abs/alg-geom/9202025v1

Mathematics

Algebraic Geometry

27 pages AmS-Tex 2.0 (plus 3 figures in LaTex)

Scientific paper

Let f : X -> S be any elliptic fibration. If X has dimension 3 and is not uniruled, then X has a minimal model (with terminal singularities) [Mori]. In earlier work we have shown that there exists a birationally equivalent elliptic fibration p: Y -> T such that Y is minimal and a multiple of K_Y can be expressed as the pullback of a divisor from T. Moreover T has at worst quotient singularities; it is not difficult to find examples where T is actually singular. In this paper we describe the singularities of this parameter surface T in the case of no multiple fibers. Although T is not uniquely determined by the birational equivalence class of the fibration, any two such T are related by a particular kind of birational map.

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