I gave this talk at the Séminaire de Géométrie Algébrique de l’Université de Lille on the 11th of September 2025 and at the Number Theory, Algebra and Geometry seminar of the University of Exeter on the 12th of November 2025.
I also gave a similar talk with a less negative focus on positive characteristic at the Algebra and Number Theory seminar of Queen Mary University of London on the 27th of March 2026.
These talks are based on the second and third sections of my thesis K3 quotients of abelian surfaces in positive characteristic (Gonzalez-Hernandez, 2025) and on my paper The classification of generalised Kummer surfaces in positive characteristic (Gonzalez-Hernandez, 2026).
References
2026
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The classification of generalised Kummer surfaces in positive characteristic
Alvaro Gonzalez-Hernandez
arXiv 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.
2025
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K3 quotients of abelian surfaces in positive characteristic
Alvaro Gonzalez-Hernandez
2025
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
