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.

arXiv comment23 pages; minor corrections and improved exposition. To appear in the proceedings of the 2025 Summer Research Institute in Algebraic Geometry