Log Contractions and Equidimensional Models of Elliptic Threefolds

This work was initially motivated by Miranda’s work on elliptic Weierstrass threefolds. An elliptic variety is a complex (irreducible and reduced) projective variety together with a morphism π : X → S whose fiber over a general point is a smooth elliptic curve. For example, if the elliptic fibration has a section, X can be locally expressed as a hypersurface [De]. This is called the Weierstrass form of the fibration; all such fibrations are (by definition) equidimensional. If dim(X) = 2, the fibration is of course always equidimensional. This is in general not true in higher dimension (see [U] for a non trivial example). Miranda [Mi] describes a smooth equidimensional (flat) model for any elliptic Weierstrass threefold; such models occur naturally in the study of moduli spaces. Each birational class of equivalent elliptic fibrations over S̄ is associated to a log variety (S̄,ΛS̄) (1.9.2)-(1.10). Here (in §2) we use Minimal model theory to link birational maps of log surfaces (log contractions) to equidimensional fibrations of elliptic threefolds. In particular we give a necessary and sufficient condition for an elliptic threefold X → S to be birationally equivalent to an equidimensional elliptic fibration X̄ → S, where X̄ has terminal singularities and S is the original basis of the fibration (Theorem 2.5). We call this an equidimensional model. As a corollary we show that an elliptic fibration of positive Kodaira dimension has a minimal model with an equidimensional birationally equivalent elliptic fibration satisfying certain good properties (Corollary 2.7) that naturally generalize the case of elliptic fibrations of surfaces. Miranda derives his result analyzing the local equation of the Weierstrass model. He shows, checking case by case, that after blowing up the surface a sufficient number of times it is possible to resolve the singularities of the threefold preserving the flatness. We present a summary of Miranda’s results in §3.