Given a finite index normal subgroup of a fundamental group, how to construct a finite etale cover over a...
$begingroup$
Let $S,C$ be smooth projective surface,curve over the complex number field $mathbb{C}$ ,respectively. Let $f:Sto C$ be a smooth fibration with general fiber $F$.Let $g=g(F)geqslant 1$ and $ngeqslant 3$. Consider the Picard-Lefschetz monodromy
$eta:pi_1(C)to Aut(H^1(F,mathbb{Z_n}))$
Since $Aut(H^1(F,mathbb{Z}_n))$ is finite, the kernel of $eta$, $Ker(eta)$ , is a normal subgroup of $pi_1(C)$ with finite index $d$. But I don't know how to construct an $acute{e}$tale cover of $C$ of degree $d$, say $widetilde{C}to C$, corresponding to the quotient group $pi_1(C)/Ker(eta)$. Once constructed, after such a base change, I want to show the pullback of $f$ will be trivial.
Any hints or references will be welcomed. Thanks
algebraic-geometry fibration
asked Dec 24 '18 at 8:13
Jiabin DuJiabin Du
301111
$endgroup$
add a comment |
$begingroup$
Let $S,C$ be smooth projective surface,curve over the complex number field $mathbb{C}$ ,respectively. Let $f:Sto C$ be a smooth fibration with general fiber $F$.Let $g=g(F)geqslant 1$ and $ngeqslant 3$. Consider the Picard-Lefschetz monodromy
$eta:pi_1(C)to Aut(H^1(F,mathbb{Z_n}))$
Since $Aut(H^1(F,mathbb{Z}_n))$ is finite, the kernel of $eta$, $Ker(eta)$ , is a normal subgroup of $pi_1(C)$ with finite index $d$. But I don't know how to construct an $acute{e}$tale cover of $C$ of degree $d$, say $widetilde{C}to C$, corresponding to the quotient group $pi_1(C)/Ker(eta)$. Once constructed, after such a base change, I want to show the pullback of $f$ will be trivial.
Any hints or references will be welcomed. Thanks
algebraic-geometry fibration
asked Dec 24 '18 at 8:13
Jiabin DuJiabin Du
301111
$endgroup$
$begingroup$
If $Gsubset pi_1(C)$ is a normal subgroup, it acts on $Y$, the universal cover of $C$ and you get a map $Y/Gto C$ which is Galois with Galois group $pi_1(C)/G$. If $G$ is of finite index in $pi_1(C)$, $Y/Gto C$ is an etale cover.
$endgroup$
– Mohan
Dec 26 '18 at 1:36
add a comment |
$begingroup$
Let $S,C$ be smooth projective surface,curve over the complex number field $mathbb{C}$ ,respectively. Let $f:Sto C$ be a smooth fibration with general fiber $F$.Let $g=g(F)geqslant 1$ and $ngeqslant 3$. Consider the Picard-Lefschetz monodromy
$eta:pi_1(C)to Aut(H^1(F,mathbb{Z_n}))$
Since $Aut(H^1(F,mathbb{Z}_n))$ is finite, the kernel of $eta$, $Ker(eta)$ , is a normal subgroup of $pi_1(C)$ with finite index $d$. But I don't know how to construct an $acute{e}$tale cover of $C$ of degree $d$, say $widetilde{C}to C$, corresponding to the quotient group $pi_1(C)/Ker(eta)$. Once constructed, after such a base change, I want to show the pullback of $f$ will be trivial.
Any hints or references will be welcomed. Thanks
algebraic-geometry fibration
asked Dec 24 '18 at 8:13
Jiabin DuJiabin Du
301111
$endgroup$
Let $S,C$ be smooth projective surface,curve over the complex number field $mathbb{C}$ ,respectively. Let $f:Sto C$ be a smooth fibration with general fiber $F$.Let $g=g(F)geqslant 1$ and $ngeqslant 3$. Consider the Picard-Lefschetz monodromy
$eta:pi_1(C)to Aut(H^1(F,mathbb{Z_n}))$
Since $Aut(H^1(F,mathbb{Z}_n))$ is finite, the kernel of $eta$, $Ker(eta)$ , is a normal subgroup of $pi_1(C)$ with finite index $d$. But I don't know how to construct an $acute{e}$tale cover of $C$ of degree $d$, say $widetilde{C}to C$, corresponding to the quotient group $pi_1(C)/Ker(eta)$. Once constructed, after such a base change, I want to show the pullback of $f$ will be trivial.
Any hints or references will be welcomed. Thanks
algebraic-geometry fibration
algebraic-geometry fibration
asked Dec 24 '18 at 8:13
Jiabin DuJiabin Du
301111
asked Dec 24 '18 at 8:13
Jiabin DuJiabin Du
301111
asked Dec 24 '18 at 8:13
Jiabin DuJiabin Du
301111
asked Dec 24 '18 at 8:13
Jiabin DuJiabin Du
301111
asked Dec 24 '18 at 8:13
Jiabin DuJiabin Du
301111
301111
$begingroup$
If $Gsubset pi_1(C)$ is a normal subgroup, it acts on $Y$, the universal cover of $C$ and you get a map $Y/Gto C$ which is Galois with Galois group $pi_1(C)/G$. If $G$ is of finite index in $pi_1(C)$, $Y/Gto C$ is an etale cover.
$endgroup$
– Mohan
Dec 26 '18 at 1:36
add a comment |
$begingroup$
If $Gsubset pi_1(C)$ is a normal subgroup, it acts on $Y$, the universal cover of $C$ and you get a map $Y/Gto C$ which is Galois with Galois group $pi_1(C)/G$. If $G$ is of finite index in $pi_1(C)$, $Y/Gto C$ is an etale cover.
$endgroup$
– Mohan
Dec 26 '18 at 1:36
$begingroup$
If $Gsubset pi_1(C)$ is a normal subgroup, it acts on $Y$, the universal cover of $C$ and you get a map $Y/Gto C$ which is Galois with Galois group $pi_1(C)/G$. If $G$ is of finite index in $pi_1(C)$, $Y/Gto C$ is an etale cover.
$endgroup$
– Mohan
Dec 26 '18 at 1:36
$begingroup$
If $Gsubset pi_1(C)$ is a normal subgroup, it acts on $Y$, the universal cover of $C$ and you get a map $Y/Gto C$ which is Galois with Galois group $pi_1(C)/G$. If $G$ is of finite index in $pi_1(C)$, $Y/Gto C$ is an etale cover.
$endgroup$
– Mohan
Dec 26 '18 at 1:36
add a comment |
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$begingroup$
If $Gsubset pi_1(C)$ is a normal subgroup, it acts on $Y$, the universal cover of $C$ and you get a map $Y/Gto C$ which is Galois with Galois group $pi_1(C)/G$. If $G$ is of finite index in $pi_1(C)$, $Y/Gto C$ is an etale cover.
$endgroup$
– Mohan
Dec 26 '18 at 1:36