We consider threefolds that admit a fibration by K3 surfaces over a nonsingular curve, equipped with a divisorial sheaf that defines a polarisation of degree two on the general fibre. Under certain assumptions on the threefold we show that its relative log canonical model exists and can be explicitly reconstructed from a small set of data determined by the original fibration. Finally we prove a converse to the above statement: under certain assumptions, any such set of data determines a threefold that arises as the relative log canonical model of a threefold admitting a fibration by K3 surfaces of degree two.
threefold, fibration, K3 surface
14J30 - $3$-folds [See also 32Q25]
14D06 - Fibrations, degenerations
14E30 - Minimal model program (Mori theory, extremal rays)
14J28 - $K3$ surfaces and Enriques surfaces