  This work was initially motivated by Miranda's work on elliptic Weierstrass
threefolds. Miranda [Mi] describes a smooth equidimensional (flat) model for
any elliptic Weierstrass threefold; such models occur naturally in the study of
moduli spaces. In this paper we use minimal model theory to link birational
maps of log surfaces (log contractions) to equidimensional fibrations of
elliptic threefolds. 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
that naturally generalize the case of elliptic fibrations of surfaces. We apply
the results to the case of Weierstrass models and give a global explanation of
Miranda's algorithm.
