Pushforwad bundle alond a degree 2 map from $P^1$ to $P^1$
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Suppose $f:P^1rightarrow P^1$ is a degree 2 morphism. Let $L$ be a line bundle on $P^1$ (which is equivalent to a $O(n)$), then what is $f_*L$? As for an open set $U$ we can see the preimage is a disjoint of two pieces of open sets, if $U$ is small enough, so the image is a rank 2 bundle. I guess that it should be $O(n)oplus O(n)$, as we can calculate the dimension of sections.
algebraic-geometry projective-space line-bundles
asked Dec 31 '18 at 9:48
Peter LiuPeter Liu
307114
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add a comment |
$begingroup$
Suppose $f:P^1rightarrow P^1$ is a degree 2 morphism. Let $L$ be a line bundle on $P^1$ (which is equivalent to a $O(n)$), then what is $f_*L$? As for an open set $U$ we can see the preimage is a disjoint of two pieces of open sets, if $U$ is small enough, so the image is a rank 2 bundle. I guess that it should be $O(n)oplus O(n)$, as we can calculate the dimension of sections.
algebraic-geometry projective-space line-bundles
asked Dec 31 '18 at 9:48
Peter LiuPeter Liu
307114
$endgroup$
add a comment |
$begingroup$
Suppose $f:P^1rightarrow P^1$ is a degree 2 morphism. Let $L$ be a line bundle on $P^1$ (which is equivalent to a $O(n)$), then what is $f_*L$? As for an open set $U$ we can see the preimage is a disjoint of two pieces of open sets, if $U$ is small enough, so the image is a rank 2 bundle. I guess that it should be $O(n)oplus O(n)$, as we can calculate the dimension of sections.
algebraic-geometry projective-space line-bundles
asked Dec 31 '18 at 9:48
Peter LiuPeter Liu
307114
$endgroup$
Suppose $f:P^1rightarrow P^1$ is a degree 2 morphism. Let $L$ be a line bundle on $P^1$ (which is equivalent to a $O(n)$), then what is $f_*L$? As for an open set $U$ we can see the preimage is a disjoint of two pieces of open sets, if $U$ is small enough, so the image is a rank 2 bundle. I guess that it should be $O(n)oplus O(n)$, as we can calculate the dimension of sections.
algebraic-geometry projective-space line-bundles
algebraic-geometry projective-space line-bundles
asked Dec 31 '18 at 9:48
Peter LiuPeter Liu
307114
asked Dec 31 '18 at 9:48
Peter LiuPeter Liu
307114
asked Dec 31 '18 at 9:48
Peter LiuPeter Liu
307114
asked Dec 31 '18 at 9:48
Peter LiuPeter Liu
307114
asked Dec 31 '18 at 9:48
Peter LiuPeter Liu
307114
307114
add a comment |
add a comment |
1 Answer
1
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votes
$begingroup$
First,
$$
f_*O cong O oplus O(-1).
$$
Indeed, this should be a rank 2 vector bundle with $H^0 = Bbbk$ and $H^1 = 0$, by Grothendieck theorem any vector bundle on $mathbb{P}^1$ is a sum of line bundles, and so the only option we have is the one written in the right hand side above. Similarly,
$$
f_*O(-1) cong O(-1) oplus O(-1).
$$
Finally, from the above and projection formula it follows that
$$
f_*O(2n) cong O(n) oplus O(n-1),
qquad
f_*O(2n-1) cong O(n-1) oplus O(n-1).
$$
answered Dec 31 '18 at 11:26
SashaSasha
5,218139
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1 Answer
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1 Answer
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votes
$begingroup$
First,
$$
f_*O cong O oplus O(-1).
$$
Indeed, this should be a rank 2 vector bundle with $H^0 = Bbbk$ and $H^1 = 0$, by Grothendieck theorem any vector bundle on $mathbb{P}^1$ is a sum of line bundles, and so the only option we have is the one written in the right hand side above. Similarly,
$$
f_*O(-1) cong O(-1) oplus O(-1).
$$
Finally, from the above and projection formula it follows that
$$
f_*O(2n) cong O(n) oplus O(n-1),
qquad
f_*O(2n-1) cong O(n-1) oplus O(n-1).
$$
answered Dec 31 '18 at 11:26
SashaSasha
5,218139
$endgroup$
add a comment |
$begingroup$
First,
$$
f_*O cong O oplus O(-1).
$$
Indeed, this should be a rank 2 vector bundle with $H^0 = Bbbk$ and $H^1 = 0$, by Grothendieck theorem any vector bundle on $mathbb{P}^1$ is a sum of line bundles, and so the only option we have is the one written in the right hand side above. Similarly,
$$
f_*O(-1) cong O(-1) oplus O(-1).
$$
Finally, from the above and projection formula it follows that
$$
f_*O(2n) cong O(n) oplus O(n-1),
qquad
f_*O(2n-1) cong O(n-1) oplus O(n-1).
$$
answered Dec 31 '18 at 11:26
SashaSasha
5,218139
$endgroup$
add a comment |
$begingroup$
First,
$$
f_*O cong O oplus O(-1).
$$
Indeed, this should be a rank 2 vector bundle with $H^0 = Bbbk$ and $H^1 = 0$, by Grothendieck theorem any vector bundle on $mathbb{P}^1$ is a sum of line bundles, and so the only option we have is the one written in the right hand side above. Similarly,
$$
f_*O(-1) cong O(-1) oplus O(-1).
$$
Finally, from the above and projection formula it follows that
$$
f_*O(2n) cong O(n) oplus O(n-1),
qquad
f_*O(2n-1) cong O(n-1) oplus O(n-1).
$$
answered Dec 31 '18 at 11:26
SashaSasha
5,218139
$endgroup$
First,
$$
f_*O cong O oplus O(-1).
$$
Indeed, this should be a rank 2 vector bundle with $H^0 = Bbbk$ and $H^1 = 0$, by Grothendieck theorem any vector bundle on $mathbb{P}^1$ is a sum of line bundles, and so the only option we have is the one written in the right hand side above. Similarly,
$$
f_*O(-1) cong O(-1) oplus O(-1).
$$
Finally, from the above and projection formula it follows that
$$
f_*O(2n) cong O(n) oplus O(n-1),
qquad
f_*O(2n-1) cong O(n-1) oplus O(n-1).
$$
answered Dec 31 '18 at 11:26
SashaSasha
5,218139
answered Dec 31 '18 at 11:26
SashaSasha
5,218139
answered Dec 31 '18 at 11:26
SashaSasha
5,218139
answered Dec 31 '18 at 11:26
SashaSasha
5,218139
5,218139
add a comment |
add a comment |
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