On the geometry of (1,2)-polarized Kummer surfaces

TL;DR

This paper explores the geometric properties of (1,2)-polarized Kummer surfaces, showing they can be represented as double covers of the projective plane branched along specific line configurations, using explicit genus-three curves.

Contribution

It provides a new geometric model for (1,2)-polarized Kummer surfaces as double covers of the plane branched along six lines, with a detailed construction involving genus-three curves.

Findings

Kummer surfaces can be modeled as double covers of the projective plane.

The branch locus consists of six lines, three of which are concurrent.

Explicit pencils of genus-three curves are used to establish the polarization.

Abstract

We discuss several geometric features of a Kummer surface associated with a (1,2)-polarized abelian surface defined over the field of complex numbers. In particular, we show that any such Kummer surface can be modeled as the double cover of the projective plane branched along six lines, three of which meet a common point. The proof uses certain explicit pencils of plane quartic bielliptic genus-three curves whose associated Prym varieties are naturally (1,2)-polarized abelian surfaces.