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# Explicit models for threefolds fibred by K3 surfaces of degree two

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• Alan Thompson,
Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, Canada, T6G 2G1
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## Abstract

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.

Keywords:  threefold, fibration, K3 surface

MSC Classifications:  14J30 - $3$-folds [See also 32Q25] 14D06 - Fibrations, degenerations 14E30 - Minimal model program (Mori theory, extremal rays) 14J28 - $K3$ surfaces and Enriques surfaces
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