Explicit construction of K3 surfaces with everywhere good reduction

2026

1. J. Symb. Comput.

   

   Explicit desingularisation of Kummer surfaces in characteristic two via specialisation

   Alvaro Gonzalez-Hernandez

   Journal of Symbolic Computation, 2026

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   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},
   }