arXiv subjects

Alvaro Gonzalez-Hernandez

Publications and source records attributed to Alvaro Gonzalez-Hernandez.

3 recordsLinked to original sources

arXiv2026

The classification of generalised Kummer surfaces in positive characteristic

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 $p\mid|G|$ 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 $p\mid|G|$, 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.

math.AG↗

arXiv2026

Intersections of the automorphism and the Ekedahl-Oort strata in $M_2$

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.

math.AG↗

arXiv2026

Explicit desingularisation of Kummer surfaces in characteristic two via specialisation

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.

math.AG↗