The Degree of Irrationality of del Pezzo Surfaces

TL;DR

This work determines the possible degrees of irrationality $\mathrm{irr}_k X$ for del Pezzo surfaces over various fields, showing the invariant lies in $\{1,2,3,4,6\}$ and is realized differently depending on the base field (finite, local, number, or arbitrary). The authors develop and deploy an irrationality criterion based on the Galois action on the geometric Picard group, leverage root-system/Weyl-group data, and analyze low- and high-degree cases separately, including degree-$1$ through degree-$9$ del Pezzo surfaces. They provide a field-by-field synthesis and, crucially, construct examples demonstrating realizability of different irrationality values (e.g., degree $9$ Brauer–Severi varieties yielding $\mathrm{irr}_k X_9=3$ when nontrivial). The results illuminate how arithmetic aspects of the base field interact with birational geometry to constrain irrationality, and they point to future work on related rational surfaces, K3s, and Fano threefolds. The study thus advances understanding of birational invariants in arithmetic geometry with concrete classification across field types.

Abstract

For an irreducible variety $X$ over a field $k$, the degree of irrationality $\operatorname{irr}_k X$ is the minimal degree of a dominant rational map $X \dashrightarrow \mathbb{P}_k^{\operatorname{\dim} X}$. When $X$ is a curve, this is simply the gonality of $X$. We determine the possible degrees of irrationality of del Pezzo surfaces over an assortment of field types: number fields, local fields, finite fields, and arbitrary fields.

Theorems & Definitions (45)

• Theorem 1.1
• Lemma 2.1
• proof
• Definition 2.2
• Remark 2.3
• Theorem 2.4
• Remark 2.5
• Remark 2.6
• Theorem 2.7: Iskovskih:Minimal-models-of-rational-surfaces-over-arbitrary-fields,VarillyAlvarado:Arithmetic-of-del-Pezzo-surfaces
• Theorem 2.8: Manin-Cubic-Forms