Finite morphisms from a curve to the projective line

Nonconstant functions are transcendental, hence define the map. Suppose first that $f$ is algebraic over $k$. By [F2] both $f$ and $f^{-1}$ are global units, so $f\in \mathrm{\Gamma }(C,{\mathcal{O}}_C{)}^{\times }$, and by [F3] this group is $k^{\times }$, so $f$ is a constant function, contrary to the hypothesis. Hence $f$ is transcendental over $k$, and the transcendental clause of [F2] provides a finite locally free morphism ${\phi }_f:C\to {\mathbb{P}}_k^1$ of degree $[k(C):k(f)]$ with ${\phi }_f^{\mathrm{\#}}(t)=f$ for the standard coordinate $t$ of ${\mathbb{P}}_k^1$.

Degree and the fibre over infinity. By [F1] the morphism ${\phi }_f$ is finite locally free of degree $[k(C):k(f)]$, its fibre over infinity is exactly the pole divisor $(f{)}_{\mathrm{\infty }}={\sum }_{{\mathrm{ord}}_x(f)<0}(-{\mathrm{ord}}_x(f))[x]$, and this divisor has degree $[k(C):k(f)]$; by [F4] the integer $[k(C):k(f)]$ is the degree of the nonconstant morphism ${\phi }_f$ in the sense of the curve-degree definition, since ${\phi }_f^{\mathrm{\#}}(t)=f$ identifies the function field of the target with $k(f)$. In particular $[k(C):k(f)]\ge 1$, a degree of a finite field extension being positive, and $(f{)}_{\mathrm{\infty }}\ge 0$ is an effective divisor.

The existence clause. Let $C$ be any smooth proper geometrically integral curve over $k$, take a closed point $p$, which exists by [F9], and an integer $n\ge 1$ with $n[\kappa (p):k]+1-g\ge 2$; by [F5] there is a nonconstant $f\in L(np)$ with $(f{)}_{\mathrm{\infty }}$ a nonzero effective divisor supported at $p$, and by steps 1.1 and 2.1 the function $f$ defines a finite locally free $k$-morphism ${\phi }_f:C\to {\mathbb{P}}_k^1$ of positive degree. Hence every such curve admits a finite $k$-morphism to the projective line, for instance one obtained from the bounded-pole corollary.

The sheaf of the pole divisor is the pullback of $\mathcal{O}(1)$. The map ${\phi }_f$ is flat by [F1], so [F7] defines the Cartier pullback. On the chart about infinity use the equation $s=1\mathrm{/}t$ for $[\mathrm{\infty }]$ and on its complement use the equation $1$. Their pullbacks are $1\mathrm{/}f$ on ${\phi }_f^{-1}(U_{\mathrm{\infty }})$ and $1$ on the complement of the infinity fibre. At a point of that fibre, $f$ has negative order, so the pulled-back equation has order $-{\mathrm{ord}}_x(f)>0$; outside the fibre the local equation is $1$ and its order is zero. Thus these are exactly the local Cartier equations of the pole divisor from [F1], and ${\phi }_f^{\ast }[\mathrm{\infty }]=(f{)}_{\mathrm{\infty }}=A$. Now [F6] and [F7] give ${\phi }_f^{\ast }{\mathcal{O}}_{{\mathbb{P}}_k^1}(1)\cong {\phi }_f^{\ast }{\mathcal{O}}_{{\mathbb{P}}_k^1}([\mathrm{\infty }])\cong {\mathcal{O}}_C({\phi }_f^{\ast }[\mathrm{\infty }])={\mathcal{O}}_C(A)\mathrm{.}$ The degree of $A$ is the weighted fibre degree in step 2.1; no claim that pullback preserves degree is needed.

Conclusion and choice accounting. Steps 1.1 and 2.1 show that every nonconstant $f\in k(C{)}^{\times }$ defines a finite locally free morphism ${\phi }_f:C\to {\mathbb{P}}_k^1$ of degree $[k(C):k(f)]\ge 1$ whose fibre over infinity is the pole divisor $(f{)}_{\mathrm{\infty }}$ of that degree; step 3.1 shows that each smooth proper geometrically integral curve over $k$ carries such a function, hence admits a finite $k$-morphism to ${\mathbb{P}}_k^1$; and step 3.2 identifies the sheaf of the pole divisor with ${\phi }_f^{\ast }{\mathcal{O}}_{{\mathbb{P}}_k^1}(1)$. The fibre-degree clause is the actual pole-map interface [F1]; the sheaf identity follows from the explicit local Cartier pullback and its canonical line-bundle isomorphism [F7]. AC is inherited through the stated suppliers as in [F8]; the existence clause uses one closed point supplied by [F9] and one function from [F5].