Explicit desingularisation of Kummer surfaces in characteristic two via specialisation

TL;DR

This paper provides explicit methods for desingularising Kummer surfaces in characteristic two, including models over various fields, and explores conditions for good reduction, with concrete examples and applications.

Contribution

It introduces explicit constructions and models for desingularised Kummer surfaces in characteristic two, extending classical descriptions and providing criteria for good reduction.

Findings

Explicit models for desingularised Kummer surfaces in characteristic two

Method to find completely desingularised models based on p-rank

Example of a Kummer surface with good reduction over a quadratic field

Abstract

We study the birational geometry of the Kummer surfaces associated to the Jacobian varieties of genus two curves, with a particular focus on fields of characteristic two. In order to do so, we explicitly compute a projective embedding of the Jacobian of a general genus two curve and, from this, we construct its associated Kummer surface. This explicit construction produces a model for desingularised Kummer surfaces over any field of characteristic not two, and specialising these equations to characteristic two provides a model of a partial desingularisation. Adapting the classic description of the Picard lattice in terms of tropes, we also describe how to explicitly find completely desingularised models of Kummer surfaces whenever the $p$-rank is not zero. In the final section of this paper, we compute an example of a Kummer surface with everywhere good reduction over a quadratic number field, and draw connections between the models we computed and a criterion that determines when a Kummer surface has good reduction at two.