---
date: 2022-02-09 11:55
modification date: Wednesday 9th February 2022 11:55:38
title: dominant morphism
aliases: [dominant rational map, dominant, dominant map, birational, birational map, rational morphism, birational morphism, birational, rational]

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Last modified date: <%+ tp.file.last_modified_date() %>

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- Tags: 
	- #AG/basics
- Refs:
	- #todo/add-references
- Links:
	- [birational geometry](Unsorted/birational%20geometry.md)
	- [resolution of singularities](Unsorted/resolution%20of%20singularities.md)

---

# rational morphisms

A **rational variety** is $X$ containing $U$ open dense with a biregular isomorphism $U\iso V \subseteq \AA^N$ for some $N$. Equivalently, $K(X)\cong K(\AA^N) = K(x_1,\cdots, x_N)$ for some $N$.

A **rational morphism** $f: X \torational Y$ 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 $U_1 \intersect U_2$.

![](attachments/Pasted%20image%2020221003223938.png)

# dominant and birational morphisms

![](attachments/Pasted%20image%2020220526135949.png)

Relation to the [rational function field](rational%20function%20field):
![](attachments/Pasted%20image%2020220526140029.png)
- Definition: a morphism $f\in \Sch(X, Y)$ is **dominant** iff $f(X) \injects Y$ is *dense*:
	- On affines, $f: \spec A \to \spec B$ has dense image iff $\ker( B\to A) \subseteq \nilrad{B}$.

![](attachments/Pasted%20image%2020220214090356.png)
![](attachments/Pasted%20image%2020220214090437.png)
![](attachments/Pasted%20image%2020220214090532.png)

A morphism $f: X\torational Y$ of irreducible varieties is **rational** iff $\im(f) \contains U$ a nonempty open subset. Note that this is precisely what is needed to define the composition of $f$ with another rational morphism $g: Y\to Z$.