Mixing
Fulton example section $2.4$, help needed in clarification
$begingroup$
This is an example in Fulton, Algebraic Curves.
Let $V= V(xw-yz) subset mathbb{A^{4}(k)}$, and $Gamma(V)$, the co-ordinate ring of $V$ is $k[x,y,z,w]/ (xw-yz)$.
and, $bar{x}, bar{y}, bar{z}, bar{w}$ denote residues of $x,y,z,w$ in $Gamma(V)$.
Then $frac{bar{x}}{bar{y}}= frac{bar{z}}{bar{w}} =f in k(V)$, (the field of rational functions on $V$) is defined at $P=(x,y,z,w) in V$ if $y,w, neq 0$
My question- I think points $P$ where f is defined should be where $bar{z}, bar{y} neq bar{0}$, the points $(x,y,z,w)$ where $y,z$ are not in $I(V)= (xw-yz)$. How come the book says otherwise? I'm confused, can someone help?
Thanks.
algebraic-geometry algebraic-curves
asked Feb 3 at 9:47
MojojojoMojojojo
1,46721131
$endgroup$
add a comment |
$begingroup$
This is an example in Fulton, Algebraic Curves.
Let $V= V(xw-yz) subset mathbb{A^{4}(k)}$, and $Gamma(V)$, the co-ordinate ring of $V$ is $k[x,y,z,w]/ (xw-yz)$.
and, $bar{x}, bar{y}, bar{z}, bar{w}$ denote residues of $x,y,z,w$ in $Gamma(V)$.
Then $frac{bar{x}}{bar{y}}= frac{bar{z}}{bar{w}} =f in k(V)$, (the field of rational functions on $V$) is defined at $P=(x,y,z,w) in V$ if $y,w, neq 0$
My question- I think points $P$ where f is defined should be where $bar{z}, bar{y} neq bar{0}$, the points $(x,y,z,w)$ where $y,z$ are not in $I(V)= (xw-yz)$. How come the book says otherwise? I'm confused, can someone help?
Thanks.
algebraic-geometry algebraic-curves
asked Feb 3 at 9:47
MojojojoMojojojo
1,46721131
$endgroup$
add a comment |
$begingroup$
This is an example in Fulton, Algebraic Curves.
Let $V= V(xw-yz) subset mathbb{A^{4}(k)}$, and $Gamma(V)$, the co-ordinate ring of $V$ is $k[x,y,z,w]/ (xw-yz)$.
and, $bar{x}, bar{y}, bar{z}, bar{w}$ denote residues of $x,y,z,w$ in $Gamma(V)$.
Then $frac{bar{x}}{bar{y}}= frac{bar{z}}{bar{w}} =f in k(V)$, (the field of rational functions on $V$) is defined at $P=(x,y,z,w) in V$ if $y,w, neq 0$
My question- I think points $P$ where f is defined should be where $bar{z}, bar{y} neq bar{0}$, the points $(x,y,z,w)$ where $y,z$ are not in $I(V)= (xw-yz)$. How come the book says otherwise? I'm confused, can someone help?
Thanks.
algebraic-geometry algebraic-curves
asked Feb 3 at 9:47
MojojojoMojojojo
1,46721131
$endgroup$
This is an example in Fulton, Algebraic Curves.
Let $V= V(xw-yz) subset mathbb{A^{4}(k)}$, and $Gamma(V)$, the co-ordinate ring of $V$ is $k[x,y,z,w]/ (xw-yz)$.
and, $bar{x}, bar{y}, bar{z}, bar{w}$ denote residues of $x,y,z,w$ in $Gamma(V)$.
Then $frac{bar{x}}{bar{y}}= frac{bar{z}}{bar{w}} =f in k(V)$, (the field of rational functions on $V$) is defined at $P=(x,y,z,w) in V$ if $y,w, neq 0$
My question- I think points $P$ where f is defined should be where $bar{z}, bar{y} neq bar{0}$, the points $(x,y,z,w)$ where $y,z$ are not in $I(V)= (xw-yz)$. How come the book says otherwise? I'm confused, can someone help?
Thanks.
algebraic-geometry algebraic-curves
algebraic-geometry algebraic-curves
asked Feb 3 at 9:47
MojojojoMojojojo
1,46721131
asked Feb 3 at 9:47
MojojojoMojojojo
1,46721131
asked Feb 3 at 9:47
MojojojoMojojojo
1,46721131
asked Feb 3 at 9:47
MojojojoMojojojo
1,46721131
asked Feb 3 at 9:47
MojojojoMojojojo
1,46721131
1,46721131
add a comment |
add a comment |
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