# Q_l-cohomology projective planes and singular Enriques surfaces in characteristic two

## TL;DR

This paper classifies singular Enriques surfaces in characteristic two with specific rational curve configurations, revealing new features and constructing related algebraic surfaces with particular cohomological properties.

## Contribution

It provides a classification of singular Enriques surfaces in characteristic two with rank nine rational curves and constructs related algebraic surfaces with ADE singularities and trivial canonical bundle.

## Key findings

- Singular Enriques surfaces in characteristic two support a rank nine configuration of rational curves.
- Construction of algebraic surfaces with ADE singularities and trivial canonical bundle matching projective plane cohomology.
- Development of existence results for classical Enriques surfaces and applications to integral models.

## Abstract

We classify singular Enriques surfaces in characteristic two supporting a rank nine configuration of smooth rational curves. They come in one-dimensional families defined over the prime field, paralleling the situation in other characteristics, but featuring novel aspects. Contracting the given rational curves, one can derive algebraic surfaces with isolated ADE-singularities and trivial canonical bundle whose Q_l-cohomology equals that of a projective plane. Similar existence results are developed for classical Enriques surfaces. We also work out an application to integral models of Enriques surfaces (and K3 surfaces).