The Degree of Irrationality of del Pezzo Surfaces

TL;DR

This paper investigates the minimal degrees of dominant rational maps from del Pezzo surfaces to projective spaces over various fields, extending understanding of their irrationality measures.

Contribution

It classifies the degrees of irrationality of del Pezzo surfaces over different types of fields, providing new insights into their birational complexity.

Findings

Determined possible degrees of irrationality over number fields.

Extended results to local and finite fields.

Provided a comprehensive classification across various 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.