On surfaces with smooth projective models over $\mathbb{Z}$

TL;DR

This paper classifies smooth projective models of surfaces with negative Kodaira dimension over rings of integers, based on their arithmetic and cohomological properties, and discusses related models over Dedekind domains.

Contribution

It provides a birational classification of such surfaces over $ ext{Spec}( ext{Z})$ and general rings of integers, linking geometric and arithmetic invariants.

Findings

Classification of models over $ ext{Spec}( ext{Z})$

Results on models over Dedekind domains

Connection between invariants and birational types

Abstract

In this expository article, we prove a birational classification of smooth projective models of surfaces with negative Kodaira dimension over $\mathbb{Z}$ and over more general rings of integers $\mathcal{O}_K$, depending on their arithmetic and cohomological invariants. Along the way we collect some results on smooth projective models of surfaces over Dedekind domains.