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# Explicit Models for Threefolds Fibred by K3 Surfaces of Degree Two

Published:2012-09-21
Printed: Aug 2013
• 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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