Is regular birational image of affine subset affine? Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 23, 2019 at 23:30 UTC (7:30pm US/Eastern)Integral extension of local rings of projective varietiesBirational equivalence between projective varieties is an equivalence relationthe definition of “Birational Equivalence”Precomposing a rational function $f$ with a birational isomorphism $g$ to make $fcirc g$ regular?Surjective regular morphism from affine space to punctured planeIs every complex affine/projective variety isomorphic/birational to one defined by an ideal $Isubset mathbbZ[x_1,dots,x_n]$?Birational affine schemesBirational $mathbbP^1$-bundlesEquivalent definition of regular map between quasi-projective varieties.what is the difference between isomorphic and birational?Is every affine variety birational to affine space?
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Is regular birational image of affine subset affine?
Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 23, 2019 at 23:30 UTC (7:30pm US/Eastern)Integral extension of local rings of projective varietiesBirational equivalence between projective varieties is an equivalence relationthe definition of “Birational Equivalence”Precomposing a rational function $f$ with a birational isomorphism $g$ to make $fcirc g$ regular?Surjective regular morphism from affine space to punctured planeIs every complex affine/projective variety isomorphic/birational to one defined by an ideal $Isubset mathbbZ[x_1,dots,x_n]$?Birational affine schemesBirational $mathbbP^1$-bundlesEquivalent definition of regular map between quasi-projective varieties.what is the difference between isomorphic and birational?Is every affine variety birational to affine space?
$begingroup$
Let $X$ and $Y$ be projective varieties and $φcolon X to Y$ a regular birational map. If $U⊆X$ is an affine subset of $X$, is $φ(U)$ affine?
This is related to my previous question.
algebraic-geometry affine-varieties
asked Apr 2 at 8:06
HumanHuman
303110
$endgroup$
add a comment |
$begingroup$
Let $X$ and $Y$ be projective varieties and $φcolon X to Y$ a regular birational map. If $U⊆X$ is an affine subset of $X$, is $φ(U)$ affine?
This is related to my previous question.
algebraic-geometry affine-varieties
asked Apr 2 at 8:06
HumanHuman
303110
$endgroup$
add a comment |
$begingroup$
Let $X$ and $Y$ be projective varieties and $φcolon X to Y$ a regular birational map. If $U⊆X$ is an affine subset of $X$, is $φ(U)$ affine?
This is related to my previous question.
algebraic-geometry affine-varieties
asked Apr 2 at 8:06
HumanHuman
303110
$endgroup$
Let $X$ and $Y$ be projective varieties and $φcolon X to Y$ a regular birational map. If $U⊆X$ is an affine subset of $X$, is $φ(U)$ affine?
This is related to my previous question.
algebraic-geometry affine-varieties
algebraic-geometry affine-varieties
asked Apr 2 at 8:06
HumanHuman
303110
asked Apr 2 at 8:06
HumanHuman
303110
asked Apr 2 at 8:06
HumanHuman
303110
asked Apr 2 at 8:06
HumanHuman
303110
asked Apr 2 at 8:06
HumanHuman
303110
303110
add a comment |
add a comment |
1 Answer
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$begingroup$
No.
The simplest example is where $varphi: X rightarrow Y$ is the blowup of a point in $mathbf P^2$. Then $X$ contains an affine open subset $U cong mathbf A^2$ and $Y$ contains an affine open subset $V cong mathbf A^2$ such that the map $varphi: U rightarrow V $ is given by the formula $(x,y) mapsto (x,xy)$. So $varphi(U) subset V cong mathbf A^2$ is the union of the open set $x neq 0$ and the point $(0,0)$. This is not affine.
answered Apr 2 at 10:32
Asal Beag DubhAsal Beag Dubh
68515
$endgroup$
add a comment |
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$begingroup$
No.
The simplest example is where $varphi: X rightarrow Y$ is the blowup of a point in $mathbf P^2$. Then $X$ contains an affine open subset $U cong mathbf A^2$ and $Y$ contains an affine open subset $V cong mathbf A^2$ such that the map $varphi: U rightarrow V $ is given by the formula $(x,y) mapsto (x,xy)$. So $varphi(U) subset V cong mathbf A^2$ is the union of the open set $x neq 0$ and the point $(0,0)$. This is not affine.
answered Apr 2 at 10:32
Asal Beag DubhAsal Beag Dubh
68515
$endgroup$
add a comment |
$begingroup$
No.
The simplest example is where $varphi: X rightarrow Y$ is the blowup of a point in $mathbf P^2$. Then $X$ contains an affine open subset $U cong mathbf A^2$ and $Y$ contains an affine open subset $V cong mathbf A^2$ such that the map $varphi: U rightarrow V $ is given by the formula $(x,y) mapsto (x,xy)$. So $varphi(U) subset V cong mathbf A^2$ is the union of the open set $x neq 0$ and the point $(0,0)$. This is not affine.
answered Apr 2 at 10:32
Asal Beag DubhAsal Beag Dubh
68515
$endgroup$
add a comment |
$begingroup$
No.
The simplest example is where $varphi: X rightarrow Y$ is the blowup of a point in $mathbf P^2$. Then $X$ contains an affine open subset $U cong mathbf A^2$ and $Y$ contains an affine open subset $V cong mathbf A^2$ such that the map $varphi: U rightarrow V $ is given by the formula $(x,y) mapsto (x,xy)$. So $varphi(U) subset V cong mathbf A^2$ is the union of the open set $x neq 0$ and the point $(0,0)$. This is not affine.
answered Apr 2 at 10:32
Asal Beag DubhAsal Beag Dubh
68515
$endgroup$
No.
The simplest example is where $varphi: X rightarrow Y$ is the blowup of a point in $mathbf P^2$. Then $X$ contains an affine open subset $U cong mathbf A^2$ and $Y$ contains an affine open subset $V cong mathbf A^2$ such that the map $varphi: U rightarrow V $ is given by the formula $(x,y) mapsto (x,xy)$. So $varphi(U) subset V cong mathbf A^2$ is the union of the open set $x neq 0$ and the point $(0,0)$. This is not affine.
answered Apr 2 at 10:32
Asal Beag DubhAsal Beag Dubh
68515
answered Apr 2 at 10:32
Asal Beag DubhAsal Beag Dubh
68515
answered Apr 2 at 10:32
Asal Beag DubhAsal Beag Dubh
68515
answered Apr 2 at 10:32
Asal Beag DubhAsal Beag Dubh
68515
68515
add a comment |
add a comment |
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