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

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.

21 pages, this is an expository article written for a volume of the Summer Research Institute in Algebraic Geometry held at Colorado State University in 2025. Comments are welcome