This talk is based on the last section of my paper Explicit desingularisation of Kummer surfaces in characteristic two via specialisation(Gonzalez-Hernandez, 2026).
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.
@article{Gonzalez-Hernandez2024ExplicitDesingularisation,title={{Explicit desingularisation of Kummer surfaces in characteristic two via specialisation}},author={Gonzalez-Hernandez, Alvaro},journal={Journal of Symbolic Computation},volume={135},pages={102541},year={2026},issn={0747-7171},doi={https://doi.org/10.1016/j.jsc.2025.102541},url={https://www.sciencedirect.com/science/article/pii/S0747717125001233},keywords={Kummer surfaces, Characteristic two, Genus two curves, Everywhere good reduction},publisher={To appear in the Journal of Symbolic Computation},eprint={2409.04532},}
