A $(\mathbb{Z}/p\mathbb{Z})^k$-cover is a flat Galois cover $X\to Y$ with Galois group $G = (\mathbb{Z}/p\mathbb{Z})^k$. In this setting, there exist several intermediate quotients $Y_1, …, Y_n$ corresponding to the subgroups $\mathbb{Z}/p\mathbb{Z} \leq G$. In this talk, we focus on the case where $Y$ is a rational surface and each $Y_i$ is either a surface with $p_g = 0$ or a K3 surface. These assumptions allow us to have strong control over the weight-2 Hodge structure of the covering surface $X$.

In particular, when $Y$ is the projective plane, we classify all covers satisfying these conditions and obtain examples of surfaces of general type $X$ with $p_g(X) > 0$. We also discuss the Infinitesimal Torelli property, Chow groups, Chow motives, and the Tate and Mumford-Tate conjectures for such surfaces $X$.

This is joint work with A. Garbagnati, and part of a subsequent collaboration with A. Ulivi and F. Fallucca.