Suppose that $\phi$ is a morphism between compleax algebraic varieties named $X$ and $Y$. I know that the degree of the morphism $\phi= [Rat(X):Rat(Y)]$. Suppose that $\phi$ is a one degree morphism.
With this assumptions is it true that $\phi$ is a 1 to 1 morphism?
2026-04-02 05:28:57.1775107737
Question about the degree of a morphism
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$\phi$ is degree one if and only if it is an isomorphism from an open subset of $X$ to an open subset of $Y$ (i.e. it is birational). So it is generically one-to-one. But birational maps can be far from injective.
As counterexamples, consider blowups of varieties along closed subvarieties. These are always birational but never injective.
What you can say is that the map $\phi$ is dominant and hence surjective if $X$ is proper.