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rational morphisms
A rational morphism \(f: X \dashrightarrow Y\) is any morphism \(\tilde f: U\to Y\) defined on some nonempty open subset \(U \subseteq X\). Two rational morphisms defined on \(U_1, U_2\) are equivalent iff they agree on \(U_1 \cap U_2\).
dominant and birational morphisms
Relation to the rational function field: 
- Definition: a morphism \(f\in {\mathsf{Sch}}(X, Y)\) is dominant iff \(f(X) \hookrightarrow Y\) is dense: - On affines, \(f: \operatorname{Spec}A \to \operatorname{Spec}B\) has dense image iff \(\ker( B\to A) \subseteq {\sqrt{0_{B}} }\).
A morphism \(f: X\dashrightarrow Y\) of irreducible varieties is rational iff \(\operatorname{im}(f) \supseteq U\) a nonempty open subset. Note that this is precisely what is needed to define the composition of \(f\) with another rational morphism \(g: Y\to Z\).
