If $I(V times W)$ is prime then $V times W$ is irreducible.
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Let $k$ be a algebraic closed field and $V subset A^m_k$ and $W subset A_k^n$ two algebraic varieties. Let $f,g in k[x_1,dots,x_m,y_1,dots,y_n]$ and $I(V)$ the ideal of $V$.
I want to show that $$Z_f ={(b_1,dots,b_n) : f(x_1,dots,x_m,b_1,dots,b_n) in I(V) }$$ is closed in $A_k^n$, but I don't have a clue how to do it.
I want to use it to prove that, if $fg in I(V times W)$, then $$W = (Z_f cap W) cup (Z_g cap W) $$ and conclude that if $I(V times W)$ is prime then $V times W$ is irreducible.
algebraic-geometry
asked Nov 27 at 12:55
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1608
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Let $k$ be a algebraic closed field and $V subset A^m_k$ and $W subset A_k^n$ two algebraic varieties. Let $f,g in k[x_1,dots,x_m,y_1,dots,y_n]$ and $I(V)$ the ideal of $V$.
I want to show that $$Z_f ={(b_1,dots,b_n) : f(x_1,dots,x_m,b_1,dots,b_n) in I(V) }$$ is closed in $A_k^n$, but I don't have a clue how to do it.
I want to use it to prove that, if $fg in I(V times W)$, then $$W = (Z_f cap W) cup (Z_g cap W) $$ and conclude that if $I(V times W)$ is prime then $V times W$ is irreducible.
algebraic-geometry
asked Nov 27 at 12:55
H R
1608
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let $k$ be a algebraic closed field and $V subset A^m_k$ and $W subset A_k^n$ two algebraic varieties. Let $f,g in k[x_1,dots,x_m,y_1,dots,y_n]$ and $I(V)$ the ideal of $V$.
I want to show that $$Z_f ={(b_1,dots,b_n) : f(x_1,dots,x_m,b_1,dots,b_n) in I(V) }$$ is closed in $A_k^n$, but I don't have a clue how to do it.
I want to use it to prove that, if $fg in I(V times W)$, then $$W = (Z_f cap W) cup (Z_g cap W) $$ and conclude that if $I(V times W)$ is prime then $V times W$ is irreducible.
algebraic-geometry
asked Nov 27 at 12:55
H R
1608
Let $k$ be a algebraic closed field and $V subset A^m_k$ and $W subset A_k^n$ two algebraic varieties. Let $f,g in k[x_1,dots,x_m,y_1,dots,y_n]$ and $I(V)$ the ideal of $V$.
I want to show that $$Z_f ={(b_1,dots,b_n) : f(x_1,dots,x_m,b_1,dots,b_n) in I(V) }$$ is closed in $A_k^n$, but I don't have a clue how to do it.
I want to use it to prove that, if $fg in I(V times W)$, then $$W = (Z_f cap W) cup (Z_g cap W) $$ and conclude that if $I(V times W)$ is prime then $V times W$ is irreducible.
algebraic-geometry
algebraic-geometry
asked Nov 27 at 12:55
H R
1608
asked Nov 27 at 12:55
H R
1608
asked Nov 27 at 12:55
H R
1608
asked Nov 27 at 12:55
H R
1608
asked Nov 27 at 12:55
H R
1608
1608
add a comment |
add a comment |
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