Research
These are my publications and theses:
2026
- The classification of generalised Kummer surfaces in positive characteristicAlvaro Gonzalez-HernandezarXiv preprint, 2026
Building on earlier work of Katsura and Rybakov, we complete the classification, in every characteristic, of all groups G acting on an abelian surface A by automorphisms preserving the group law such that the resolution of the quotient A/G is a K3 surface. In order to do so, we study actions of groups with p∣|G| in characteristics p=2,3 and 5. Using the theory of rational double points in positive characteristic, we show that the possible ADE singularity types of A/G are constrained by the requirement that their local fundamental group contains G as a subgroup, and we determine the singular locus of A/G via the action of G on the \ell-adic Tate module of A. As a key step in the classification, we prove that if A is a supersingular abelian surface and p∣|G|, then A/G can never be a generalised Kummer surface. Finally, we construct explicit examples of such generalised Kummer surfaces as quotients of products of two elliptic curves.
- Generalised Kummer surfaces in positive characteristic, Séminaire de Géométrie Algébrique de l'Université de Lille (September 2025)
- Crazy for three, K3 quotients of abelian surfaces by a group of order three in characteristic three, Young Researchers in Algebraic Number Theory (Y-RANT) (September 2025)
2025
- K3 quotients of abelian surfaces in positive characteristicAlvaro Gonzalez-Hernandez2025
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
- Generalised Kummer surfaces of Jacobians of genus two curves, Arithmetic, Geometry and Computations on K3 surfaces (September 2025)
- Generalised Kummer surfaces in positive characteristic, Séminaire de Géométrie Algébrique de l'Université de Lille (September 2025)
- Constructing K3 surfaces as quotients of abelian surfaces, London Junior Number Theory Seminar (May 2025)
- Intersections of the automorphism and the Ekedahl-Oort strata in the moduli space of genus two curvesAlvaro Gonzalez-HernandezarXiv preprint, 2025
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.
- Intersections of the automorphism and p-rank strata inside \(M_2\), VII Congreso de Jóvenes Investigadores de la RSME (January 2025)
2024
- Explicit desingularisation of Kummer surfaces in characteristic two via specialisationAlvaro Gonzalez-HernandezJournal of Symbolic Computation (2026), arXiv preprint, 2024
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.
- K3 surfaces with everywhere good reduction, Young Researchers in Algebraic Number Theory (July 2024)
- Explicit construction of K3 surfaces with everywhere good reduction, Oberseminar on Arithmetische Geometrie (June 2024)
- Explicit models of Kummer surfaces in characteristic two, 7th Symposium of the Roman Number Theory Association (May 2024)
- How to desingularise Kummer surfaces, Junior Geometry Seminar of KCL/UCL (February 2024)
- Explicit theory of Kummer surfaces in characteristic two, Hodge Club (February 2024)
- Crazy for two, genus two curves in characteristic two via Kummer surfaces, Young Researchers in Algebraic Number Theory (Y-RANT) (September 2023)
- Kummer surfaces, bridging the gap between geometry and number theory, Queer in Number Theory and Geometry (August 2023)
2022
2021
- Local fields and the Hasse principleAlvaro Gonzalez-HernandezUniversity of Oxford, 2021
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.
- An introduction to the Hasse principle through examples, Warwick's Postgraduate Seminar (January 2022)
2020
- Continued fractionsAlvaro Gonzalez-HernandezUniversidad de Salamanca, 2020
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.
