Gonality

Let $k$ be a field and let $C$ be a smooth proper geometrically integral curve over $k$ (Curves over a field). The gonality of $C$ is expressed by the raw minimum $\mathrm{gon}(C):=\min \{\mathrm{deg}(\phi ):\phi :C\to {\mathbb{P}}_k^1\text{ is a nonconstant }k\text{-morphism}\},$ where degree is as in Degree of a nonconstant morphism of curves. Under AC, the set in this expression is nonempty and the minimum exists, as follows.

Assume AC. The function field $k(C)$ has transcendence degree one over $k$ (Function field of an integral finite-type scheme, Affine-domain dimension equals transcendence degree). Hence there is an $f\in k(C)$ transcendental over $k$. In particular $f\mathrm{\ne }0$ and is nonconstant. The actual finite-map result A nonconstant rational function defines a finite map to the projective line, whose Statement assumes AC, produces a finite locally free nonconstant morphism ${\phi }_f:C\to {\mathbb{P}}_k^1$ of degree $[k(C):k(f)]$. Thus the set of degrees in the display is nonempty. Every such degree is a positive integer (Degree of a nonconstant morphism of curves); well-ordering of the positive integers gives a least element. Therefore the displayed minimum exists under AC and is a positive integer.

Under the same AC assumption, $\mathrm{gon}(C)=1$ if and only if $C\cong {\mathbb{P}}_k^1$. If the minimum is $1$, it is attained by a nonconstant morphism $\phi :C\to {\mathbb{P}}_k^1$ of degree $1$. By the degree definition, the induced finite extension of function fields has degree one, so $\phi$ is birational; the actual birational-smooth-proper-curve theorem Birational smooth proper curves are isomorphic then makes it an isomorphism. Conversely, an isomorphism $C\to {\mathbb{P}}_k^1$ has degree one, and every nonconstant curve-map degree is positive, so its gonality is one. The AC hypotheses here are inherited from the cited finite-map and birational-curve suppliers; the raw minimum notation itself adds no choice principle.