## Rocky Mountain Journal of Mathematics

### Kummer surfaces and K3 surfaces with $(\mathbb{Z} /2\mathbb{Z} )^4$ symplectic action

#### Abstract

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.

#### Article information

Source
Rocky Mountain J. Math., Volume 46, Number 4 (2016), 1141-1205.

Dates
First available in Project Euclid: 19 October 2016

https://projecteuclid.org/euclid.rmjm/1476884579

Digital Object Identifier
doi:10.1216/RMJ-2016-46-4-1141

Mathematical Reviews number (MathSciNet)
MR3563178

Zentralblatt MATH identifier
1370.14033

#### Citation

Garbagnati, Alice; Sarti, Alessandra. Kummer surfaces and K3 surfaces with $(\mathbb{Z} /2\mathbb{Z} )^4$ symplectic action. Rocky Mountain J. Math. 46 (2016), no. 4, 1141--1205. doi:10.1216/RMJ-2016-46-4-1141. https://projecteuclid.org/euclid.rmjm/1476884579

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