Restricting divisors to closed fiber of relative curve over henselian DVR
$begingroup$
Setup:
$k$ is an algebraically closed field.
$mathcal{O} = k{t}$ is the henselization of $k[t]_{(t)}$.
$V rightarrow text{Spec}(mathcal{O})$ is proper and has a section.
$V$ is irreducible, nonsingular, and of dimension $2$.
$X$ is the closed fiber of $V$.
Question:
Why is $text{ker}(text{Pic}(V) rightarrow text{Pic}(X))$ uniquely divisible by $n$ when $n$ is prime to $text{char}(k)$?
This is claimed (no proof) in Artin, Grothendieck Topologies, Prop. 4.4.2.
algebraic-geometry
asked Dec 21 '18 at 17:10
rj7k8rj7k8
671212
$endgroup$
add a comment |
$begingroup$
Setup:
$k$ is an algebraically closed field.
$mathcal{O} = k{t}$ is the henselization of $k[t]_{(t)}$.
$V rightarrow text{Spec}(mathcal{O})$ is proper and has a section.
$V$ is irreducible, nonsingular, and of dimension $2$.
$X$ is the closed fiber of $V$.
Question:
Why is $text{ker}(text{Pic}(V) rightarrow text{Pic}(X))$ uniquely divisible by $n$ when $n$ is prime to $text{char}(k)$?
This is claimed (no proof) in Artin, Grothendieck Topologies, Prop. 4.4.2.
algebraic-geometry
asked Dec 21 '18 at 17:10
rj7k8rj7k8
671212
$endgroup$
add a comment |
$begingroup$
Setup:
$k$ is an algebraically closed field.
$mathcal{O} = k{t}$ is the henselization of $k[t]_{(t)}$.
$V rightarrow text{Spec}(mathcal{O})$ is proper and has a section.
$V$ is irreducible, nonsingular, and of dimension $2$.
$X$ is the closed fiber of $V$.
Question:
Why is $text{ker}(text{Pic}(V) rightarrow text{Pic}(X))$ uniquely divisible by $n$ when $n$ is prime to $text{char}(k)$?
This is claimed (no proof) in Artin, Grothendieck Topologies, Prop. 4.4.2.
algebraic-geometry
asked Dec 21 '18 at 17:10
rj7k8rj7k8
671212
$endgroup$
Setup:
$k$ is an algebraically closed field.
$mathcal{O} = k{t}$ is the henselization of $k[t]_{(t)}$.
$V rightarrow text{Spec}(mathcal{O})$ is proper and has a section.
$V$ is irreducible, nonsingular, and of dimension $2$.
$X$ is the closed fiber of $V$.
Question:
Why is $text{ker}(text{Pic}(V) rightarrow text{Pic}(X))$ uniquely divisible by $n$ when $n$ is prime to $text{char}(k)$?
This is claimed (no proof) in Artin, Grothendieck Topologies, Prop. 4.4.2.
algebraic-geometry
algebraic-geometry
asked Dec 21 '18 at 17:10
rj7k8rj7k8
671212
asked Dec 21 '18 at 17:10
rj7k8rj7k8
671212
asked Dec 21 '18 at 17:10
rj7k8rj7k8
671212
asked Dec 21 '18 at 17:10
rj7k8rj7k8
671212
asked Dec 21 '18 at 17:10
rj7k8rj7k8
671212
671212
add a comment |
add a comment |
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