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.<\/p>\n
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<\/a>. 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 \u2229 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<\/a>. Finally, an integral morphism of finite type is finite.<\/p>\n Of course, the same thing is true for proper morphisms… see Lemma Tag 02UP<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":" 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 … Continue reading