The Degree of Irrationality of del Pezzo Surfaces

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.