Mixing
Cokernel of morphism defined by a section
$begingroup$
Let $X$ be a smooth projective curve over the field of complex numbers. Let $V$ be a vector bundle on $X$ of rank $r$. Suppose $sin H^0(X,V)$ then we have an injective morphism given by
$$0rightarrow O_Xrightarrow V,.$$
What is the cokernel of the above morphism? If $V$ is rank two is it given by $text{det }V$?
algebraic-geometry
asked Jan 8 at 13:15
user52991user52991
336310
$endgroup$
add a comment |
$begingroup$
Let $X$ be a smooth projective curve over the field of complex numbers. Let $V$ be a vector bundle on $X$ of rank $r$. Suppose $sin H^0(X,V)$ then we have an injective morphism given by
$$0rightarrow O_Xrightarrow V,.$$
What is the cokernel of the above morphism? If $V$ is rank two is it given by $text{det }V$?
algebraic-geometry
asked Jan 8 at 13:15
user52991user52991
336310
$endgroup$
$begingroup$
What about the inclusion of $mathcal{O}_X$ in $mathcal{O}_X(P)$, where $P$ is a point on $X$?
$endgroup$
– Ariyan Javanpeykar
Jan 8 at 13:33
$begingroup$
@Ariyan, I know that the cokernel that case is a skyscraper sheaf supported at $P$. I am interested in the case when rank is two, which I have already mentioned in the question
$endgroup$
– user52991
Jan 8 at 14:32
$begingroup$
For the cokernel to be a vector bundle, one need that $s$ does not vanish.
$endgroup$
– Roland
Jan 8 at 15:25
add a comment |
$begingroup$
Let $X$ be a smooth projective curve over the field of complex numbers. Let $V$ be a vector bundle on $X$ of rank $r$. Suppose $sin H^0(X,V)$ then we have an injective morphism given by
$$0rightarrow O_Xrightarrow V,.$$
What is the cokernel of the above morphism? If $V$ is rank two is it given by $text{det }V$?
algebraic-geometry
asked Jan 8 at 13:15
user52991user52991
336310
$endgroup$
Let $X$ be a smooth projective curve over the field of complex numbers. Let $V$ be a vector bundle on $X$ of rank $r$. Suppose $sin H^0(X,V)$ then we have an injective morphism given by
$$0rightarrow O_Xrightarrow V,.$$
What is the cokernel of the above morphism? If $V$ is rank two is it given by $text{det }V$?
algebraic-geometry
algebraic-geometry
asked Jan 8 at 13:15
user52991user52991
336310
asked Jan 8 at 13:15
user52991user52991
336310
asked Jan 8 at 13:15
user52991user52991
336310
asked Jan 8 at 13:15
user52991user52991
336310
asked Jan 8 at 13:15
user52991user52991
336310
336310
$begingroup$
What about the inclusion of $mathcal{O}_X$ in $mathcal{O}_X(P)$, where $P$ is a point on $X$?
$endgroup$
– Ariyan Javanpeykar
Jan 8 at 13:33
$begingroup$
@Ariyan, I know that the cokernel that case is a skyscraper sheaf supported at $P$. I am interested in the case when rank is two, which I have already mentioned in the question
$endgroup$
– user52991
Jan 8 at 14:32
$begingroup$
For the cokernel to be a vector bundle, one need that $s$ does not vanish.
$endgroup$
– Roland
Jan 8 at 15:25
add a comment |
$begingroup$
What about the inclusion of $mathcal{O}_X$ in $mathcal{O}_X(P)$, where $P$ is a point on $X$?
$endgroup$
– Ariyan Javanpeykar
Jan 8 at 13:33
$begingroup$
@Ariyan, I know that the cokernel that case is a skyscraper sheaf supported at $P$. I am interested in the case when rank is two, which I have already mentioned in the question
$endgroup$
– user52991
Jan 8 at 14:32
$begingroup$
For the cokernel to be a vector bundle, one need that $s$ does not vanish.
$endgroup$
– Roland
Jan 8 at 15:25
$begingroup$
What about the inclusion of $mathcal{O}_X$ in $mathcal{O}_X(P)$, where $P$ is a point on $X$?
$endgroup$
– Ariyan Javanpeykar
Jan 8 at 13:33
$begingroup$
What about the inclusion of $mathcal{O}_X$ in $mathcal{O}_X(P)$, where $P$ is a point on $X$?
$endgroup$
– Ariyan Javanpeykar
Jan 8 at 13:33
$begingroup$
@Ariyan, I know that the cokernel that case is a skyscraper sheaf supported at $P$. I am interested in the case when rank is two, which I have already mentioned in the question
$endgroup$
– user52991
Jan 8 at 14:32
$begingroup$
@Ariyan, I know that the cokernel that case is a skyscraper sheaf supported at $P$. I am interested in the case when rank is two, which I have already mentioned in the question
$endgroup$
– user52991
Jan 8 at 14:32
$begingroup$
For the cokernel to be a vector bundle, one need that $s$ does not vanish.
$endgroup$
– Roland
Jan 8 at 15:25
$begingroup$
For the cokernel to be a vector bundle, one need that $s$ does not vanish.
$endgroup$
– Roland
Jan 8 at 15:25
add a comment |
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$begingroup$
What about the inclusion of $mathcal{O}_X$ in $mathcal{O}_X(P)$, where $P$ is a point on $X$?
$endgroup$
– Ariyan Javanpeykar
Jan 8 at 13:33
$begingroup$
@Ariyan, I know that the cokernel that case is a skyscraper sheaf supported at $P$. I am interested in the case when rank is two, which I have already mentioned in the question
$endgroup$
– user52991
Jan 8 at 14:32
$begingroup$
For the cokernel to be a vector bundle, one need that $s$ does not vanish.
$endgroup$
– Roland
Jan 8 at 15:25