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Degree $deg(f)$ of a Fibration
Clash Royale CLAN TAG#URR8PPP
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Let $S$ be a surface (so a $2$ dimensional, proper $k$-scheme) and $B$ a curve (" $1$ dim " ).
Consider a finite fibration $f:S to B$ (so we have $mathcalO_B = f_*mathcalO_S $).
My question is how to see that $deg(f)=0$?
Here the definition of degree according to wiki: $$deg(f):= [K(S):K(B)]$$
where $K(S), K(B)$ are function fields of $S$ and $B$.
Seemingly we must have $deg(f)= infty$ since $S$ is a surface, while $B$ a curve by considering the transcendental degrees of function field extension.
Background: Show that $(K_S cdot K_S)=0$ using bundle formula containing pullbacks $f^*(K_B otimes mathcalL)$ for an appropriate invertible sheaf $mathcalL$ and using the fact
$$(f^*mathcalG cdot f^*mathcalG) = deg(f)(mathcalG cdot mathcalG)$$
algebraic-geometry surfaces schemes
asked Aug 2 at 19:02
KarlPeter
495313
add a comment |Â
up vote
-1
down vote
favorite
Let $S$ be a surface (so a $2$ dimensional, proper $k$-scheme) and $B$ a curve (" $1$ dim " ).
Consider a finite fibration $f:S to B$ (so we have $mathcalO_B = f_*mathcalO_S $).
My question is how to see that $deg(f)=0$?
Here the definition of degree according to wiki: $$deg(f):= [K(S):K(B)]$$
where $K(S), K(B)$ are function fields of $S$ and $B$.
Seemingly we must have $deg(f)= infty$ since $S$ is a surface, while $B$ a curve by considering the transcendental degrees of function field extension.
Background: Show that $(K_S cdot K_S)=0$ using bundle formula containing pullbacks $f^*(K_B otimes mathcalL)$ for an appropriate invertible sheaf $mathcalL$ and using the fact
$$(f^*mathcalG cdot f^*mathcalG) = deg(f)(mathcalG cdot mathcalG)$$
algebraic-geometry surfaces schemes
asked Aug 2 at 19:02
KarlPeter
495313
1
There is no reason in general for $(K_Scdot K_S)=0$ without further assumptions. For example, take $S$ to be the blown up of the projective plane at a point. The you have a morphism $f:Sto mathbbP^1$, satisfying your conditions, but $K_S^2=8$.
– Mohan
Aug 2 at 22:05
@Mohan: Would it work if one consider an elliptic fibration, therefore the fiber of generic point $eta in B$ is elliptic curve $S_eta$ (so with genug $g=1$)?
– KarlPeter
Aug 3 at 7:12
add a comment |Â
up vote
-1
down vote
favorite
up vote
-1
down vote
favorite
Let $S$ be a surface (so a $2$ dimensional, proper $k$-scheme) and $B$ a curve (" $1$ dim " ).
Consider a finite fibration $f:S to B$ (so we have $mathcalO_B = f_*mathcalO_S $).
My question is how to see that $deg(f)=0$?
Here the definition of degree according to wiki: $$deg(f):= [K(S):K(B)]$$
where $K(S), K(B)$ are function fields of $S$ and $B$.
Seemingly we must have $deg(f)= infty$ since $S$ is a surface, while $B$ a curve by considering the transcendental degrees of function field extension.
Background: Show that $(K_S cdot K_S)=0$ using bundle formula containing pullbacks $f^*(K_B otimes mathcalL)$ for an appropriate invertible sheaf $mathcalL$ and using the fact
$$(f^*mathcalG cdot f^*mathcalG) = deg(f)(mathcalG cdot mathcalG)$$
algebraic-geometry surfaces schemes
asked Aug 2 at 19:02
KarlPeter
495313
Let $S$ be a surface (so a $2$ dimensional, proper $k$-scheme) and $B$ a curve (" $1$ dim " ).
Consider a finite fibration $f:S to B$ (so we have $mathcalO_B = f_*mathcalO_S $).
My question is how to see that $deg(f)=0$?
Here the definition of degree according to wiki: $$deg(f):= [K(S):K(B)]$$
where $K(S), K(B)$ are function fields of $S$ and $B$.
Seemingly we must have $deg(f)= infty$ since $S$ is a surface, while $B$ a curve by considering the transcendental degrees of function field extension.
Background: Show that $(K_S cdot K_S)=0$ using bundle formula containing pullbacks $f^*(K_B otimes mathcalL)$ for an appropriate invertible sheaf $mathcalL$ and using the fact
$$(f^*mathcalG cdot f^*mathcalG) = deg(f)(mathcalG cdot mathcalG)$$
algebraic-geometry surfaces schemes
asked Aug 2 at 19:02
KarlPeter
495313
edited Aug 2 at 19:20
asked Aug 2 at 19:02
KarlPeter
495313
asked Aug 2 at 19:02
KarlPeter
495313
asked Aug 2 at 19:02
KarlPeter
495313
495313
1
There is no reason in general for $(K_Scdot K_S)=0$ without further assumptions. For example, take $S$ to be the blown up of the projective plane at a point. The you have a morphism $f:Sto mathbbP^1$, satisfying your conditions, but $K_S^2=8$.
– Mohan
Aug 2 at 22:05
@Mohan: Would it work if one consider an elliptic fibration, therefore the fiber of generic point $eta in B$ is elliptic curve $S_eta$ (so with genug $g=1$)?
– KarlPeter
Aug 3 at 7:12
add a comment |Â
1
There is no reason in general for $(K_Scdot K_S)=0$ without further assumptions. For example, take $S$ to be the blown up of the projective plane at a point. The you have a morphism $f:Sto mathbbP^1$, satisfying your conditions, but $K_S^2=8$.
– Mohan
Aug 2 at 22:05
@Mohan: Would it work if one consider an elliptic fibration, therefore the fiber of generic point $eta in B$ is elliptic curve $S_eta$ (so with genug $g=1$)?
– KarlPeter
Aug 3 at 7:12
1
1
There is no reason in general for $(K_Scdot K_S)=0$ without further assumptions. For example, take $S$ to be the blown up of the projective plane at a point. The you have a morphism $f:Sto mathbbP^1$, satisfying your conditions, but $K_S^2=8$.
– Mohan
Aug 2 at 22:05
There is no reason in general for $(K_Scdot K_S)=0$ without further assumptions. For example, take $S$ to be the blown up of the projective plane at a point. The you have a morphism $f:Sto mathbbP^1$, satisfying your conditions, but $K_S^2=8$.
– Mohan
Aug 2 at 22:05
@Mohan: Would it work if one consider an elliptic fibration, therefore the fiber of generic point $eta in B$ is elliptic curve $S_eta$ (so with genug $g=1$)?
– KarlPeter
Aug 3 at 7:12
@Mohan: Would it work if one consider an elliptic fibration, therefore the fiber of generic point $eta in B$ is elliptic curve $S_eta$ (so with genug $g=1$)?
– KarlPeter
Aug 3 at 7:12
add a comment |Â
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1
There is no reason in general for $(K_Scdot K_S)=0$ without further assumptions. For example, take $S$ to be the blown up of the projective plane at a point. The you have a morphism $f:Sto mathbbP^1$, satisfying your conditions, but $K_S^2=8$.
– Mohan
Aug 2 at 22:05
@Mohan: Would it work if one consider an elliptic fibration, therefore the fiber of generic point $eta in B$ is elliptic curve $S_eta$ (so with genug $g=1$)?
– KarlPeter
Aug 3 at 7:12