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

TL;DR

The paper addresses the problem of classifying smooth projective schemes over $\mathbb{Z}$, focusing on surfaces of negative Kodaira dimension, by combining birational MMP arguments with biregular models over Dedekind bases. It develops a framework for arithmetic MMP on families over Dedekind schemes (no flips in the smooth setting), derives a concrete classification of end products as either $\mathbb{P}^2$- or conic-bundle structures, and provides explicit treatments of Hirzebruch surfaces via elementary transformations and their deformation behavior. The authors establish detailed structure results for Picard schemes and curves over $\mathbb{Z}$, prove that smooth models of Hirzebruch surfaces over Dedekind bases can be connected by elementary transformations, and finally classify smooth del Pezzo surfaces over $\mathbb{Z}$, including explicit blow-up models and general-position constraints. Together, these results yield a concrete, Kodaira-dimension-driven panorama of how smooth projective surfaces can sit over $\mathbb{Z}$ and more general Dedekind rings, with explicit geometric and arithmetic constraints and several natural open questions remaining in cases Murphy (2) and (3) of the main theorem.

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.

Theorems & Definitions (76)

• Theorem 1.1: See \ref{['thm: rough_classification_Z']}
• Proposition 1
• Remark 1
• Remark 2
• Remark 3
• Example 1
• Theorem 2.1: Minkowski
• Theorem 2.2: Grothendieck
• Corollary 1
• proof