# Enriques surfaces with normal K3-like coverings

## TL;DR

This paper studies special types of Enriques surfaces in characteristic two with normal K3-like coverings, introducing new methods to construct and analyze their singularities, including cases with rational surface coverings and elliptic double points.

## Contribution

It develops general construction techniques for Enriques surfaces with normal K3-like coverings, including cases with rational surfaces, and introduces a theory of Zariski singularities and canonical coverings.

## Key findings

- Elliptic double points occur on these surfaces.
- A method using flops on Frobenius pullbacks constructs desired coverings.
- Conditions for tangent sheaves to be locally free are identified.

## Abstract

We analyze the structure of simply-connected Enriques surface in characteristic two whose K3-like covering is normal, building on the work of Ekedahl, Hyland and Shepherd-Barron. We develop general methods to construct such surfaces and the resulting twistor lines in the moduli stack of Enriques surfaces, including the case that the K3-like covering is a normal rational surface rather then a normal K3 surface. Among other things, we show that elliptic double points indeed do occur. In this case,there is only one singularity.The main idea is to apply flops to Frobenius pullbacks of rational elliptic surfaces, to get the desired K3-like covering. Our results hinge on Lang's classification of rational elliptic surfaces, the determination of their Mordell--Weil lattices by Shioda and Oguiso, and the behavior of unstable fibers under Frobenius pullback via Ogg's Formula. Along the way, we develop a general theory of Zariski singularities in arbitrary dimension, which is tightly interwoven with the theory of height-one group schemes actions and restricted Lie algebras. Furthermore, we determine under what conditions tangent sheaves are locally free, and introduce a theory of canonical coverings for arbitrary proper algebraic schemes.