Explicit desingularisation of Kummer surfaces in characteristic two via specialisation

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.

30 pages. Code added as an ancillary file. Accepted version to be published in the Journal of Symbolic Computation

References in corpus (6)

• On good reduction of some K3 surfaces related to abelian surfaces
• Examples of abelian surfaces with everywhere good reduction
• Degeneration of Kummer surfaces
• Two-coverings of Jacobians of curves of genus two
• Kummer Surfaces, Isogenies and Theta Functions
• Reduction of Kummer surfaces modulo 2 in the non-supersingular case