These are some of the projects that I have written or I am in the process of writing.
Currently, I am looking at how to construct projective embeddings for generalised Kummer surfaces, which are K3 surfaces that are quotients of abelian surfaces by the action of an automorphism group. My goal is to use these embeddings to study which singularities we obtain when the characteristic of the base field divides the order of the automorphism.
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 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},year={2024},pages={34},publisher={arXiv},eprint={2409.04532},}
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.
