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={To appear in the 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},}
In this thesis, we study classical and generalised Kummer surfaces, with a particular focus on the case of positive characteristic. We extend the current classification of generalised Kummer surfaces to characteristics two, three, and five, and construct many explicit examples, both when the abelian surface is the product of two elliptic curves and when it is the Jacobian of a genus two curve. We also prove several related results, including the existence of Kummer surfaces with everywhere good reduction over number fields, and a description of the intersections between the automorphism strata and the Ekedahl–Oort strata inside the moduli space of genus two curves
We compute the intersections between the automorphism strata and the pullback by the Torelli map of the Ekedahl-Oort strata inside the moduli space of genus two curves. We first describe explicitly which possible automorphism groups a genus two curve can have over a field of positive characteristic, and parametrise the families of curves with a prescribed automorphism group. Then, we describe an algorithm to compute the strata of genus two curves whose Jacobian variety has a fixed Ekedahl-Oort type. Finally, we compute the dimension and number of irreducible components of the intersections between the strata.
@article{Gonzalez-Hernandez2025IntersectionsStrata,title={{Intersections of the automorphism and the Ekedahl-Oort strata in the moduli space of genus two curves}},author={Gonzalez-Hernandez, Alvaro},year={2025},pages={21},publisher={arXiv},eprint={2507.07278},journal={arXiv preprint. Comments are welcome!}}
This is a survey of the theory of elliptic K3 surfaces and Mordell-Weil lattices. As an application of this theory, we explain how to compute bounds for the Picard number of an elliptic surface.
We discuss the main results of the theory of local fields and how it allows us to compute the ranks of elliptic curves through the method of complete 2-descent. The knowledge gained from this theory is then used to analyse a family of counterexamples to the Hasse principle depending on a prime parameter. This family of curves arises as the homogeneous spaces that appear when applying complete 2-descent to an elliptic curve with non-trivial Tate-Shafarevich group.
Continued fractions have played an important role in the development of many mathematical theories and, even today, they are still a very active line of research. The study of continued fractions with complex coefficients allows us to define meromorphic functions as continued fractions from their formal power series with the help of a sequence of rational approximations known as Padé approximants. Furthermore, there is an equivalence between real numbers and simple continued fractions (a special case of continued fractions with integer coefficients) and, based on this equivalence, we study problems of number theory such as how well irrational numbers can be approximated by rational numbers or how to solve Pell’s equation.
