On Enriques Surfaces with Four Cusps

Abstract

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).