# \'{E}tale coverings in codimension 1 with applications to Mori Dream Spaces

## TL;DR

This paper explores the relationship between étale coverings in codimension 1 and fundamental groups to better understand the topology of Mori dream spaces, providing explicit coverings and conditions for their properties.

## Contribution

It introduces new canonical étale coverings in codimension 1 for Mori dream spaces and toric varieties, and establishes conditions under which these coverings retain Mori dream space properties.

## Key findings

- Explicit universal étale coverings for toric varieties.
- Canonical Galois étale coverings for Mori dream spaces.
- Conditions ensuring coverings are Mori dream spaces.

## Abstract

The present paper is devoted to developing relations between Galois \'etale coverings in codimension 1 and \'etale fundamental groups in codimension 1 of algebraic varieties, aimed to studying the topology of Mori dream spaces. In particular, the universal \'etale covering in codimension 1 of a non-degenerate toric variety and a canonical Galois \'etale covering in codimension 1 of a Mori dream space (MDS) are exhibited. Sufficient conditions for the latter being either still a MDS or the universal \'etale covering in codimension 1 are given. As an application, a canonical toric embedding of K3 universal coverings, of Enriques surfaces which are Mori dream, is described.