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We present a systematic study of threefolds fibred by K3 surfaces that aremirror to sextic double planes. There are many parallels between this theoryand the theory of elliptic surfaces. We show that the geometry of suchthreefolds is controlled by a pair of invariants, called the generalizedfunctional and generalized homological invariants, and we derive an explicitbirational model for them, which we call the Weierstrass form. We then describehow to resolve the singularities of the Weierstrass form to obtain the "minimalform", which has mild singularities and is unique up to birational maps incodimension 2. Finally we describe some of the geometric properties ofthreefolds in minimal form, including their singular fibres, canonical divisor,and Betti numbers.

Threefolds Fibred by Mirror Sextic Double Planes