Symplectic Automorphisms on Kummer Surfaces

TL;DR

This paper investigates symplectic automorphisms on Kummer surfaces, exploring cases where the discriminant of the fixed sublattice differs from previously known examples, and clarifies the underlying reasons.

Contribution

It identifies specific Kummer surfaces exhibiting discrepancies in discriminants and explains the reasons behind these differences in the context of symplectic automorphisms.

Findings

Discovered Kummer surfaces with unique discriminant properties

Explained the discrepancy in discriminants for certain automorphism groups

Extended understanding of symplectic automorphisms on K3 surfaces

Abstract

Nikulin proved that the isometries induced on the second cohomology group of a K3 surface $X$ by a finite abelian group $G$ of symplectic automorphisms are essentially unique. Moreover he computed the discriminant of the sublattice of $H^2(X, \Z)$ which is fixed by the isometries induced by $G$. However for certain groups these discriminants are not the same of those found for explicit examples. Here we describe Kummer surfaces for which this phenomena happens and we explain the difference.