Geometric gonality (reviewed)

Let $X$ be a geometrically integral algebraic curve over a field $k$. Let $\overline{k}$ be an algebraic closure of $k$. Let $X_{\overline{k}}$ be the base extension. The geometric gonality (or $\overline{k}$-gonality) of $X$ is the gonality of $X_{\overline{k}}$, that is, the minimal degree of a dominant morphism $X_{\overline{k}} \to \mathbb P^1_{\overline{k}}$.