Suppose that f : X —> Y is a morphism of projective varieties and y is a point of Y such that there are only finitely many points x_1, …, x_r in X mapping to y. Then there exists an affine open neighborhood V of y in Y such that f^{-1}(V) —> V is finite.

How do you prove this? Here is a fun argument. First you prove that f is a projective morphism, and hence we can generalize the statement to arbitrary projective morphism. This is good because then we can localize on Y and reach the situation where Y is affine. In this case X is quasi-projective and we can find an affine open U of X containing x_1, …, x_r, see Lemma Tag 01ZY. Then f(X \ U) is closed and does not contain y. Hence we can find a principal open V of Y such that f^{-1}(V) \subset U. In particular f^{-1}(V) = U ∩ f^{-1}(V) is a principal open of U, whence affine. Now f^{-1}(V) —> V is a projective morphism of affines. There is a cute argument proving that a universally closed morphism of affines is an integral morphism, see Lemma Tag 01WM. Finally, an integral morphism of finite type is finite.

Of course, the same thing is true for proper morphisms… see Lemma Tag 02UP.