Using resultants to show extension of function fields of curves is algebraic.
$begingroup$
We are given two irreducible nonsingular plane curves, the zero sets of $f,gin bar{k}[x,y]$, with $bar{k}$ algebraically closed. We have an injective map given on algebras as $phi^*:C_frightarrow C_g$, and we want to show the the associated field extension $Quot(C_f)rightarrow Quot(C_g)$ is algebraic.
The problem is to express that $x,yin C_g$ are algebraic over $Quot(C_f)$, using resultants. The hint given is to use the resultants $Res_x(phi^*(u),phi^*(v))$ and $Res_y(phi^*(u),phi^*(v))$, where we are using $u,v$ to be $x,y$, but relabelling to avoid overloading notation, and emphasise they are coming from $C_f$.
I dont have a good feeling for how to prove things like this using resultants, they still seem a bit mysterious to me, so any additional intuition for how to think about resultants would be very welcome.
Note, I am aware of other ways to do this problem, I am specifically looking for a resultant based solution.
algebraic-geometry algebraic-curves resultant
asked Dec 13 '18 at 10:06
user277182user277182
556212
$endgroup$
add a comment |
$begingroup$
We are given two irreducible nonsingular plane curves, the zero sets of $f,gin bar{k}[x,y]$, with $bar{k}$ algebraically closed. We have an injective map given on algebras as $phi^*:C_frightarrow C_g$, and we want to show the the associated field extension $Quot(C_f)rightarrow Quot(C_g)$ is algebraic.
The problem is to express that $x,yin C_g$ are algebraic over $Quot(C_f)$, using resultants. The hint given is to use the resultants $Res_x(phi^*(u),phi^*(v))$ and $Res_y(phi^*(u),phi^*(v))$, where we are using $u,v$ to be $x,y$, but relabelling to avoid overloading notation, and emphasise they are coming from $C_f$.
I dont have a good feeling for how to prove things like this using resultants, they still seem a bit mysterious to me, so any additional intuition for how to think about resultants would be very welcome.
Note, I am aware of other ways to do this problem, I am specifically looking for a resultant based solution.
algebraic-geometry algebraic-curves resultant
asked Dec 13 '18 at 10:06
user277182user277182
556212
$endgroup$
add a comment |
$begingroup$
We are given two irreducible nonsingular plane curves, the zero sets of $f,gin bar{k}[x,y]$, with $bar{k}$ algebraically closed. We have an injective map given on algebras as $phi^*:C_frightarrow C_g$, and we want to show the the associated field extension $Quot(C_f)rightarrow Quot(C_g)$ is algebraic.
The problem is to express that $x,yin C_g$ are algebraic over $Quot(C_f)$, using resultants. The hint given is to use the resultants $Res_x(phi^*(u),phi^*(v))$ and $Res_y(phi^*(u),phi^*(v))$, where we are using $u,v$ to be $x,y$, but relabelling to avoid overloading notation, and emphasise they are coming from $C_f$.
I dont have a good feeling for how to prove things like this using resultants, they still seem a bit mysterious to me, so any additional intuition for how to think about resultants would be very welcome.
Note, I am aware of other ways to do this problem, I am specifically looking for a resultant based solution.
algebraic-geometry algebraic-curves resultant
asked Dec 13 '18 at 10:06
user277182user277182
556212
$endgroup$
We are given two irreducible nonsingular plane curves, the zero sets of $f,gin bar{k}[x,y]$, with $bar{k}$ algebraically closed. We have an injective map given on algebras as $phi^*:C_frightarrow C_g$, and we want to show the the associated field extension $Quot(C_f)rightarrow Quot(C_g)$ is algebraic.
The problem is to express that $x,yin C_g$ are algebraic over $Quot(C_f)$, using resultants. The hint given is to use the resultants $Res_x(phi^*(u),phi^*(v))$ and $Res_y(phi^*(u),phi^*(v))$, where we are using $u,v$ to be $x,y$, but relabelling to avoid overloading notation, and emphasise they are coming from $C_f$.
I dont have a good feeling for how to prove things like this using resultants, they still seem a bit mysterious to me, so any additional intuition for how to think about resultants would be very welcome.
Note, I am aware of other ways to do this problem, I am specifically looking for a resultant based solution.
algebraic-geometry algebraic-curves resultant
algebraic-geometry algebraic-curves resultant
asked Dec 13 '18 at 10:06
user277182user277182
556212
asked Dec 13 '18 at 10:06
user277182user277182
556212
asked Dec 13 '18 at 10:06
user277182user277182
556212
asked Dec 13 '18 at 10:06
user277182user277182
556212
asked Dec 13 '18 at 10:06
user277182user277182
556212
556212
add a comment |
add a comment |
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