On the generalised Kummer fourfold of the Jacobian of a genus two curve

TL;DR

This paper constructs a birational model of the generalized Kummer fourfold associated with a genus two curve's Jacobian, revealing its geometric properties and singularities, and establishing a finite ramified cover to projective space.

Contribution

It provides a new geometric interpretation and explicit birational model of the generalized Kummer fourfold for genus two curves, including analysis of its singularities and coverings.

Findings

Model has mild singularities

Admits a finite ramified cover to P^4

Based on properties of cubics on the curve

Abstract

We construct a birational model of the generalised Kummer fourfold of the Jacobian of a genus two curve, based on a geometric interpretation of the addition law on this Jacobian, obtained by the properties of the linear system of cubics on that curve. We show that our model has mild singularities and that it admits a finite ramified covering to the four-dimensional projective space.