在代数几何 中，有理映射 是定义在概形 的稠密开集上的态射。有理映射及由此引生的双有理等价 是古典代数几何学的主要对象。
固定概形
V
,
W
{\displaystyle V,W}
(
U
,
f
)
{\displaystyle (U,f)}
U
⊂
V
{\displaystyle U\subset V}
f
:
U
→
W
{\displaystyle f:U\to W}
U
{\displaystyle U}
U
{\displaystyle U}
f
{\displaystyle f}
(
U
,
f
)
∼
(
U
′
,
g
)
⟺
f
|
U
∩
U
′
=
g
|
U
∩
U
′
{\displaystyle (U,f)\sim (U',g)\iff f|_{U\cap U'}=g|_{U\cap U'}}
此外，注意到稠密性保证
U
∩
U
′
{\displaystyle U\cap U'}
V
{\displaystyle V}
V
{\displaystyle V}
V
{\displaystyle V}
W
{\displaystyle W}
分离概形 ，则任一等价类有唯一一个定义域最大的代表元。
从概形
V
{\displaystyle V}
W
{\displaystyle W}
有理映射
f
{\displaystyle f}
[
U
,
f
]
{\displaystyle [U,f]}
若
f
{\displaystyle f}
U
{\displaystyle U}
V
{\displaystyle V}
g
{\displaystyle g}
V
{\displaystyle V}
W
{\displaystyle W}
g
∘
f
{\displaystyle g\circ f}
f
{\displaystyle f}
(
U
0
,
f
U
0
)
{\displaystyle (U_{0},f_{U_{0}})}
V
{\displaystyle V}
g
{\displaystyle g}
(
V
0
,
g
V
0
)
{\displaystyle (V_{0},g_{V_{0}})}
f
U
0
(
U
0
)
∩
V
0
{\displaystyle f_{U_{0}}(U_{0})\cap V_{0}}
g
∘
f
:=
[
f
U
0
−
1
(
V
0
)
,
g
V
0
∘
f
U
0
]
{\displaystyle g\circ f:=[f_{U_{0}}^{-1}(V_{0}),g_{V_{0}}\circ f_{U_{0}}]}
同理，若
V
{\displaystyle V}
W
{\displaystyle W}
S
{\displaystyle S}
S
{\displaystyle S}
设
k
{\displaystyle k}
整环 ，设
V
:=
A
k
n
{\displaystyle V:=\mathbb {A} _{k}^{n}}
W
:=
A
k
m
{\displaystyle W:=\mathbb {A} _{k}^{m}}
V
{\displaystyle V}
W
{\displaystyle W}
f
{\displaystyle f}
f
=
(
f
1
(
x
1
,
…
,
x
n
)
g
1
(
x
1
,
…
,
x
n
)
,
…
,
f
m
(
x
1
,
…
,
x
n
)
g
m
(
x
1
,
…
,
x
n
)
)
{\displaystyle f=\left({\dfrac {f_{1}(x_{1},\ldots ,x_{n})}{g_{1}(x_{1},\ldots ,x_{n})}},\ldots ,{\dfrac {f_{m}(x_{1},\ldots ,x_{n})}{g_{m}(x_{1},\ldots ,x_{n})}}\right)}
其中
f
i
,
g
i
{\displaystyle f_{i},g_{i}}
A
k
n
∖
⋃
i
{
g
i
=
0
}
{\displaystyle \mathbb {A} _{k}^{n}\setminus \bigcup _{i}\{g_{i}=0\}}
此外，对于不可约
k
{\displaystyle k}
X
{\displaystyle X}
X
{\displaystyle X}
P
k
1
{\displaystyle \mathbb {P} _{k}^{1}}
Grothendieck, Alexandre; Jean Dieudonné. Éléments de géométrie algébrique 2nd edition. Berlin; New York: Springer-Verlag. 1971. ISBN 978-3-540-05113-8 （法语） . Hartshorne, Robin. Algebraic Geoemtry. Berlin; New York: Springer-Verlag. 1977. ISBN 978-0-387-90244-9 （英语） . 
![{\displaystyle [U,f]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6a91475f1b336cf76afe510cfe1825f4210a77aa)
![{\displaystyle g\circ f:=[f_{U_{0}}^{-1}(V_{0}),g_{V_{0}}\circ f_{U_{0}}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e837c0b956f9460195768cb054e9ab5edcc2845e)
![{\displaystyle k[x,y,z]/(xy-z)\simeq k[x,y]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ae0fa419da14a5e654fa4e05a85626c8e2691442)
