K3 surfaces with everywhere good reduction

Young Researchers in Algebraic Number Theory

I have given this talk twice:

In the Décimas Jornadas de Teoría de Números, which took place at La Corrala (Madrid) on the 10th of July of 2024.
In the conference Young Researchers in Algebraic Number Theory (Y-RANT), which took place at the University of Oxford on the 31st of July of 2024.

This talk is based on the last section of my paper Explicit desingularisation of Kummer surfaces in characteristic two via specialisation (Gonzalez-Hernandez, 2024).

References

2024

J. Symb. Comput.

Explicit desingularisation of Kummer surfaces in characteristic two via specialisation

Alvaro Gonzalez-Hernandez

Journal of Symbolic Computation (2026), arXiv preprint, 2024

Abs HTML

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.