Classification of SNC $(p,k)$-planes with Hodge structures of K3 type

A SNC $(p,k)$-plane is a $(\mathbb{Z}\mathrm{/}p{)}^k$-Galois cover $\pi \text{\hspace{0.17em}⁣}:X\to {\mathbb{P}}^2$ with a simple normal crossing branch locus. In this setting, there exist several intermediate quotients $Y_1,Y_2,\dots ,Y_{{\phi }_{p,k}}$ corresponding to subgroups of $(\mathbb{Z}\mathrm{/}p{)}^k$ of index $p$. We focus on the case where each $Y_i$ is either a surface with geometric genus $0$ or $1$, with at least one with geometric genus $1$. We classify all SNC $(p,k)$-planes satisfying these conditions obtaining that an extensive list of cases are minimal regular surfaces of general type with positive geometric genus. Moreover, these assumptions allow us to have strong control over the weight-$2$ Hodge structure of the minimal model of the covering surface $X$. Therefore, we also prove that Tate and Mumford-Tate conjectures hold for such surfaces. As an independent result of our methods, we obtain a classification of SNC $(p,1)$-planes with geometric genus $1$, which turn out to be precisely all K3 surfaces, and a classification of SNC $(p,1)$-planes with geometric genus $0$, which turn out to be instead rational surfaces.