Let $C$ be a curve over $k$ and $f:C \to \mathbb{P}^2 _k$ a morphism such that $\mathcal{L}=f^*\mathcal{O}_{\mathbb{P}^2}(1)$ is ample, therefore sets of the shape $X_s$ for global section $s$ form a topological base.
My question is why does here ampleness of $\mathcal{L}$ imply that $f$ is a finite morphism, therefore
$\mathbb{P}^2$ has an open cover by affine schemes $V_i = Spec B_i$ such that for each $i$ , $f ^{− 1}(V_i) = U_i$ is an open affine subscheme $U_i= Spec A_i$ and the induced morphisms $ B_i \to A_i $ make $A_i$ a finitely generated module over $B_i$.
My attempts: It suffies to show that $f ^{− 1}(D(T_i)) = f ^{− 1}(Spec \ k[T_j/T_i]_{j=1,2,3})$ is affine (so $= Spec(A_i)$ and $A_i$ is a finite generated $k[T_j/T_i]_{j=1,2,3}$ -module, because $D(T_i)$ cover $\mathbb{P}^2$...