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2016 Kummer surfaces and K3 surfaces with $(\mathbb{Z} /2\mathbb{Z} )^4$ symplectic action
Alice Garbagnati, Alessandra Sarti
Rocky Mountain J. Math. 46(4): 1141-1205 (2016). DOI: 10.1216/RMJ-2016-46-4-1141


In the first part of this paper we give a survey of classical results on Kummer surfaces with Picard number~17 from the point of view of lattice theory. We prove ampleness properties for certain divisors on Kummer surfaces, and we use them to describe projective models of Kummer surfaces of $(1,d)$-polarized abelian surfaces for $d=1,2,3$. As a consequence, we prove that, in these cases, the N\'eron-Severi group can be generated by lines.

In the second part of the paper we use Kummer surfaces to obtain results on K3 surfaces with a symplectic action of the group $(\mathbb{Z} /2\mathbb{Z} )^4$. In particular, we describe the possible N\'eron-Severi groups of the latter in the case that the Picard number is $16$, which is the minimal possible. We also describe the N\'eron-Severi groups of the minimal resolution of the quotient surfaces which have 15 nodes. We extend certain classical results on Kummer surfaces to these families.


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Alice Garbagnati. Alessandra Sarti. "Kummer surfaces and K3 surfaces with $(\mathbb{Z} /2\mathbb{Z} )^4$ symplectic action." Rocky Mountain J. Math. 46 (4) 1141 - 1205, 2016.


Published: 2016
First available in Project Euclid: 19 October 2016

zbMATH: 1370.14033
MathSciNet: MR3563178
Digital Object Identifier: 10.1216/RMJ-2016-46-4-1141

Primary: 14J28
Secondary: 14J10 , 14J50

Keywords: automorphisms , Enriques involutions , even sets of nodes , K3 surfaces , Kummer surfaces

Rights: Copyright © 2016 Rocky Mountain Mathematics Consortium


Vol.46 • No. 4 • 2016
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