Assuem we have a finite surjective map between two irreducible, separated schemes, $f:X \rightarrow Y$, and for a dense open $U \subset Y$ and for any $y \in U$, $|X\_y| =1$, then can we say $f$ is injective everywhere?


We can also assume that $Y$ is normal.



Injective map between two schemes
With some additional hypotheses this follows from Zariski's Main theorem.



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> If $f \colon X \to Y$ is a birational projective morphism between noetherian integral schemes, then the inverse image of every normal point of $Y$ is connected. [Hartshorne (1977, Corollary III.11.4)]
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See also: <https://en.wikipedia.org/wiki/Zariski%27s_main_theorem>




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If you make the assumption that $X$ and $Y$ are noetherian and integral, I guess you will be there.


