Log contractions and equidimensional models of elliptic threefolds

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.