The Coble-Mukai lattice from $\mathbb Q$-Gorenstein deformations

TL;DR

This paper explores the geometric properties of Enriques surfaces through $ ext{Q}$-Gorenstein smoothings of Coble surfaces, identifying their lattices and discussing applications to Gorenstein $ ext{Q}$-Homology projective planes.

Contribution

It explicitly identifies the Enriques lattice with the Coble-Mukai lattice in the context of $ ext{Q}$-Gorenstein smoothings, providing new insights into their geometric relationship.

Findings

Identification of the Enriques lattice with the Coble-Mukai lattice.

Explicit description of geometric properties of Enriques surfaces.

Applications to Gorenstein $ ext{Q}$-Homology projective planes.

Abstract

We show some geometric properties of Enriques surfaces via $\mathbb Q$-Gorenstein smoothings of Coble surfaces. In particular, we explicitly identify the Enriques lattice of the general fiber with the Coble-Mukai lattice. At the end, we discuss applications to Gorenstein $\mathbb Q$-Homology projective planes with trivial canonical class.