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Two polarised K3 surfaces associated to the same cubic fourfold

Part of: Surfaces and higher-dimensional varieties Birational geometry

Published online by Cambridge University Press:  16 March 2020

EMMA BRAKKEE
EMMA BRAKKEE*
Affiliation:
Korteweg–de Vries Institute, University of Amsterdam, P.O. Box 94248, 1090 GEAmsterdam, Netherlands. e-mail: [email protected]
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Abstract

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For infinitely many d, Hassett showed that special cubic fourfolds of discriminant d are related to polarised K3 surfaces of degree d via their Hodge structures. For half of the d, each associated K3 surface (S, L) canonically yields another one, (Sτ, Lτ). We prove that Sτ is isomorphic to the moduli space of stable coherent sheaves on S with Mukai vector (3, L, d/6). We also explain for which d the Hilbert schemes Hilbn (S) and Hilbn (Sτ) are birational.

MSC classification

Primary: 14J28: $K3$ surfaces and Enriques surfaces
Secondary: 14J10: Families, moduli, classification 14E05: Rational and birational maps
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 171 , Issue 1 , July 2021 , pp. 51 - 64
Creative Commons
Creative Common License - CCCreative Common License - BYCreative Common License - NCCreative Common License - SA
This is an Open Access article, distributed under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike licence (http://creativecommons.org/licenses/by-nc-sa/4.0/), which permits non-commercial re-use, distribution, and reproduction in any medium, provided the same Creative Commons licence is included and the original work is properly cited. The written permission of Cambridge University Press must be obtained for commercial re-use.
Copyright
© Cambridge Philosophical Society 2020

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