Proper scheme such that every vector bundle is trivial Announcing the arrival of Valued...
Proper scheme such that every vector bundle is trivial
Announcing the arrival of Valued Associate #679: Cesar Manara
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It is claimed here that there exist proper schemes (probably over a field but not explicitly stated) with trivial Picard group. This means that every locally free $O_X$-module of rank 1 is trivial.
Do there exist proper schemes over a field such that every locally free $O_X$-module of finite rank is trivial?
Maybe it is easier to give some examples with algebraic spaces, but the accepted answer should give a scheme.
EDIT: examples should be positive-dimensional.
ag.algebraic-geometry vector-bundles picard-group
asked yesterday
schematic_boischematic_boi
2137
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add a comment |
$begingroup$
It is claimed here that there exist proper schemes (probably over a field but not explicitly stated) with trivial Picard group. This means that every locally free $O_X$-module of rank 1 is trivial.
Do there exist proper schemes over a field such that every locally free $O_X$-module of finite rank is trivial?
Maybe it is easier to give some examples with algebraic spaces, but the accepted answer should give a scheme.
EDIT: examples should be positive-dimensional.
ag.algebraic-geometry vector-bundles picard-group
asked yesterday
schematic_boischematic_boi
2137
New contributor
schematic_boi is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
add a comment |
$begingroup$
It is claimed here that there exist proper schemes (probably over a field but not explicitly stated) with trivial Picard group. This means that every locally free $O_X$-module of rank 1 is trivial.
Do there exist proper schemes over a field such that every locally free $O_X$-module of finite rank is trivial?
Maybe it is easier to give some examples with algebraic spaces, but the accepted answer should give a scheme.
EDIT: examples should be positive-dimensional.
ag.algebraic-geometry vector-bundles picard-group
asked yesterday
schematic_boischematic_boi
2137
New contributor
schematic_boi is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
It is claimed here that there exist proper schemes (probably over a field but not explicitly stated) with trivial Picard group. This means that every locally free $O_X$-module of rank 1 is trivial.
Do there exist proper schemes over a field such that every locally free $O_X$-module of finite rank is trivial?
Maybe it is easier to give some examples with algebraic spaces, but the accepted answer should give a scheme.
EDIT: examples should be positive-dimensional.
ag.algebraic-geometry vector-bundles picard-group
ag.algebraic-geometry vector-bundles picard-group
asked yesterday
schematic_boischematic_boi
2137
New contributor
schematic_boi is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked yesterday
schematic_boischematic_boi
2137
New contributor
schematic_boi is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited yesterday
schematic_boi
asked yesterday
schematic_boischematic_boi
2137
New contributor
schematic_boi is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked yesterday
schematic_boischematic_boi
2137
asked yesterday
schematic_boischematic_boi
2137
2137
New contributor
schematic_boi is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
schematic_boi is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
schematic_boi is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
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1 Answer
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According to this paper it is not known if any proper algebraic scheme admits nontrivial vector bundles. Partial results can be found here.
answered yesterday
Mere ScribeMere Scribe
8192819
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great! the paper looks like it is pretty old. Hopefully, @SándorKovács could add something useful.
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– schematic_boi
yesterday
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1 Answer
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active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
According to this paper it is not known if any proper algebraic scheme admits nontrivial vector bundles. Partial results can be found here.
answered yesterday
Mere ScribeMere Scribe
8192819
$endgroup$
$begingroup$
great! the paper looks like it is pretty old. Hopefully, @SándorKovács could add something useful.
$endgroup$
– schematic_boi
yesterday
add a comment |
$begingroup$
According to this paper it is not known if any proper algebraic scheme admits nontrivial vector bundles. Partial results can be found here.
answered yesterday
Mere ScribeMere Scribe
8192819
$endgroup$
$begingroup$
great! the paper looks like it is pretty old. Hopefully, @SándorKovács could add something useful.
$endgroup$
– schematic_boi
yesterday
add a comment |
$begingroup$
According to this paper it is not known if any proper algebraic scheme admits nontrivial vector bundles. Partial results can be found here.
answered yesterday
Mere ScribeMere Scribe
8192819
$endgroup$
According to this paper it is not known if any proper algebraic scheme admits nontrivial vector bundles. Partial results can be found here.
answered yesterday
Mere ScribeMere Scribe
8192819
answered yesterday
Mere ScribeMere Scribe
8192819
answered yesterday
Mere ScribeMere Scribe
8192819
answered yesterday
Mere ScribeMere Scribe
8192819
8192819
$begingroup$
great! the paper looks like it is pretty old. Hopefully, @SándorKovács could add something useful.
$endgroup$
– schematic_boi
yesterday
add a comment |
$begingroup$
great! the paper looks like it is pretty old. Hopefully, @SándorKovács could add something useful.
$endgroup$
– schematic_boi
yesterday
$begingroup$
great! the paper looks like it is pretty old. Hopefully, @SándorKovács could add something useful.
$endgroup$
– schematic_boi
yesterday
$begingroup$
great! the paper looks like it is pretty old. Hopefully, @SándorKovács could add something useful.
$endgroup$
– schematic_boi
yesterday
add a comment |
schematic_boi is a new contributor. Be nice, and check out our Code of Conduct.
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