The Degree of Irrationality of del Pezzo Surfaces

For an irreducible variety $X$ over a field $k$, the degree of irrationality ${\mathrm{irr}}_{k}X$ is the minimal degree of a dominant rational map $X⇢{\mathbb{P}}_k^{\mathrm{}X}$. 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.