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ORBIT PARAMETRIZATIONS FOR K3 SURFACES

Part of: Surfaces and higher-dimensional varieties

Published online by Cambridge University Press:  07 July 2016

MANJUL BHARGAVA ,
WEI HO  and
ABHINAV KUMAR
MANJUL BHARGAVA
Affiliation:
Department of Mathematics, Princeton University, Princeton, NJ 08544, USA; bhargava@math.princeton.edu
WEI HO
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA; weiho@umich.edu
ABHINAV KUMAR
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Department of Mathematics, Stony Brook University, Stony Brook, NY 11794, USA; thenav@gmail.com

Abstract

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We study moduli spaces of lattice-polarized K3 surfaces in terms of orbits of representations of algebraic groups. In particular, over an algebraically closed field of characteristic 0, we show that in many cases, the nondegenerate orbits of a representation are in bijection with K3 surfaces (up to suitable equivalence) whose Néron–Severi lattice contains a given lattice. An immediate consequence is that the corresponding moduli spaces of these lattice-polarized K3 surfaces are all unirational. Our constructions also produce many fixed-point-free automorphisms of positive entropy on K3 surfaces in various families associated to these representations, giving a natural extension of recent work of Oguiso.

MSC classification

Primary: 14J28: $K3$ surfaces and Enriques surfaces 14J10: Families, moduli, classification 14J50: Automorphisms of surfaces and higher-dimensional varieties
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 4 , 2016 , e18
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2016

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