Rocky Mountain Journal of Mathematics

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

Alice Garbagnati and Alessandra Sarti

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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

Permanent link to this document
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

Subjects
Primary: 14J28: $K3$ surfaces and Enriques surfaces
Secondary: 14J10: Families, moduli, classification: algebraic theory 14J50: Automorphisms of surfaces and higher-dimensional varieties

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

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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