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

Fabio Bernasconi; Gebhard Martin; Zsolt Patakfalvi

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

Algebraic Geometry 2026-01-21 v1

Authors: Fabio Bernasconi , Gebhard Martin , Zsolt Patakfalvi

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.

Keywords

del pezzo surfaces algebraic surfaces surface theory

Cite

@article{arxiv.2601.13277,
  title  = {On surfaces with smooth projective models over $\mathbb{Z}$},
  author = {Fabio Bernasconi and Gebhard Martin and Zsolt Patakfalvi},
  journal= {arXiv preprint arXiv:2601.13277},
  year   = {2026}
}

Comments

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

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

Abstract

Keywords

Cite

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