Gonality of a curve (awaiting review)

The gonality (or $k$-gonality) of an integral algebraic curve $X$ over a field $k$ is the minimal degree of a dominant $k$-morphism $X \to \mathbb P^1_k$. Equivalently, it is the minimum of $[k(X):k(t)]$ as $t$ ranges over (transcendental) elements of the function field $k(X)$ of $X$.

The base extension of any dominant morphism $X \to \mathbb{P}^1_k$ is a dominant morphism $X_{\overline{k}} \to \mathbb{P}^1_{\overline{k}}$, so the geometric gonality of $X$ is less than or equal to the $k$-gonality of $X$.