Exercise 111.34.12. Let $k$ be an algebraically closed field. Let $f : X \to Y$ be a morphism of projective varieties such that $f^{-1}(\{ y\} )$ is finite for every closed point $y \in Y$. Prove that $f$ is finite as a morphism of schemes. Hints: (a) being finite is a local property, (b) try to reduce to Exercise 111.34.11, and (c) use a closed immersion $X \to \mathbf{P}^ n_ k$ to get a closed immersion $X \to \mathbf{P}^ n_ Y$ over $Y$.

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