# Publications of the Research Institute for Mathematical Sciences

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**Volume 54, Issue 3, 2018, pp. 433–468**

**DOI: 10.4171/PRIMS/54-3-1**

Published online: 2018-07-23

On Enriques Surfaces with Four Cusps

Sławomir Rams^{[1]}and Matthias Schütt

^{[2]}(1) Jagiellonian University, Kraków, Poland

(2) Leibniz-Universität Hannover, Germany

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).

*Keywords: *Enriques surface, cusp, three-divisible set, fundamental group, elliptic fibration, lattice polarization, K3 surface

Rams Sławomir, Schütt Matthias: On Enriques Surfaces with Four Cusps. *Publ. Res. Inst. Math. Sci.* 54 (2018), 433-468. doi: 10.4171/PRIMS/54-3-1