On Enriques Surfaces with Four Cusps

We study Enriques surfaces with four disjoint A$_2$-configurations. In particular, we construct open Enriques surfaces with fundamental groups $(\mathbb Z/3\mathbb Z)^{\oplus 2} \times \mathbb Z/2\mathbb Z$ and $\mathbb Z/6\mathbb Z$, completing the picture of the A$_2$-case from Keum and Zhang (Fundamental groups of open K3 surfaces, Enriques surfaces and Fano 3-folds, J. Pure Appl. Algebra 170 (2002), 67–91; Zbl 1060.14057). We also construct an explicit Gorenstein $\mathbb Q$-homology projective plane of singularity type A$_3+3$A$_2$, supporting an open case from Hwang, Keum and Ohashi (Gorenstein $\mathbb Q$-homology projective planes, Science China Mathematics 58 (2015), 501–512; Zbl 1314.14072).