Problem 8.17 Let $X, Y$ be nonsingular projectives curves, $ f:X\to Y $ a dominating morphism. Prove that $ f (X)=Y $.
Im trying to solve the problem 8.17 of Algebraic Curves Book of Fulton, there is a hint "If $P \in Y \setminus f(X)$, then $\widetilde{f}(\Gamma(Y\setminus\{P\}))\subset\Gamma(X)=k$", the part i don't understand is $\Gamma(X)=k$, Why?
Perhaps the rational functions defined on a nonsingular variety is isomorphic to a field , so the evaluation homomorphism and that .. ? or why?
Help me! Please... Thanks! (:
On any projective variety there are no nonconstant functions, so $\Gamma(X)$ just consists of constants and is therefore isomorphic to $k$.