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Threefolds fibred by mirror sextic double planes

Part of: Families, fibrations Surfaces and higher-dimensional varieties

Published online by Cambridge University Press:  24 June 2020

Remkes Kooistra  and
Alan Thompson
Remkes Kooistra
Affiliation:
The King’s University, 9125 – 50 St NW, Edmonton, AB T6B 2H3, Canada e-mail: Remkes.Kooistra@kingsu.ca
Alan Thompson*
Affiliation:
Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire LE11 3TU, United Kingdom
*
e-mail: A.M.Thompson@lboro.ac.uk
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Abstract

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

Keywords

K3 surfacethreefoldfibration

MSC classification

Primary: 14D06: Fibrations, degenerations
Secondary: 14J30: $3$-folds 14J28: $K3$ surfaces and Enriques surfaces
Type
Article
Information
Canadian Journal of Mathematics , Volume 73 , Issue 5 , October 2021 , pp. 1305 - 1346
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Copyright
© Canadian Mathematical Society 2020

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