# Explicit Nikulin configurations on Kummer surfaces

## TL;DR

This paper explores explicit constructions of non-isomorphic Kummer structures on Kummer surfaces, extending previous methods to broader cases and addressing limitations in earlier approaches.

## Contribution

It generalizes explicit constructions of non-isomorphic Kummer structures on Kummer surfaces with weaker polarization restrictions.

## Key findings

- Constructed explicit non-isomorphic Kummer structures on certain Kummer surfaces.
- Extended previous constructions to cases with lower polarization degrees.
- Identified scenarios where earlier methods are not applicable.

## Abstract

A Nikulin configuration is the data of $16$ disjoint smooth rational curves on a K3 surface. According to results of Nikulin, the existence of a Nikulin configuration means that the K3 surface is a Kummer surface, moreover the abelian surface from the Kummer structure is determined by the $16$ curves. A classical question of Shioda is about the existence of non isomorphic Kummer structures on the same Kummer K3 surface. The question was studied by several authors, and it was shown that the number of non-isomorphic Kummer structures is finite, but no explicit geometric construction of such structures was given. In a previous paper, we constructed explicitly non isomorphic Kummer structures on some Kummer surfaces. In this paper we generalise the construction to Kummer surfaces with a weaker restriction on the degree of the polarization and we describe some cases where the previous construction does not work.