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Proper scheme such that every vector bundle is trivial
Announcing the arrival of Valued Associate #679: Cesar Manara
Unicorn Meta Zoo #1: Why another podcast?Non-projective smooth complete threefolds with a pair of points intersecting every surfacePicard groups of non-projective varietiesVector bundle and inverse imageFlat quasi-coherent modules as colimits of locally free onesPicard Group of Projective Bundle over an Integral schemeDo cohomologically trivial line bundles affect morphisms?Characterization of schemes whose dualizing complex is perfectAre the global sections of a vector bundle a projective module? Modules over an Azumaya algebra and modules over the associated Brauer-Severi varietyIs there a group scheme G such that there is a proper class of non-isomorphic fpqc G-bundles over some fixed base?Witten's construction of vector bundle over Teichmüller space
$begingroup$
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 10 hours ago
schematic_boischematic_boi
1637
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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 10 hours ago
schematic_boischematic_boi
1637
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 10 hours ago
schematic_boischematic_boi
1637
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 10 hours ago
schematic_boischematic_boi
1637
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 10 hours ago
schematic_boischematic_boi
1637
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 7 hours ago
schematic_boi
asked 10 hours ago
schematic_boischematic_boi
1637
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 10 hours ago
schematic_boischematic_boi
1637
asked 10 hours ago
schematic_boischematic_boi
1637
1637
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 7 hours ago
Mere ScribeMere Scribe
8192819
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$begingroup$
great! the paper looks like it is pretty old. Hopefully, @SándorKovács could add something useful.
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– schematic_boi
7 hours ago
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1 Answer
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1 Answer
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active
oldest
votes
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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 7 hours ago
Mere ScribeMere Scribe
8192819
$endgroup$
$begingroup$
great! the paper looks like it is pretty old. Hopefully, @SándorKovács could add something useful.
$endgroup$
– schematic_boi
7 hours ago
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 7 hours ago
Mere ScribeMere Scribe
8192819
$endgroup$
$begingroup$
great! the paper looks like it is pretty old. Hopefully, @SándorKovács could add something useful.
$endgroup$
– schematic_boi
7 hours ago
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 7 hours ago
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 7 hours ago
Mere ScribeMere Scribe
8192819
answered 7 hours ago
Mere ScribeMere Scribe
8192819
answered 7 hours ago
Mere ScribeMere Scribe
8192819
answered 7 hours ago
Mere ScribeMere Scribe
8192819
8192819
$begingroup$
great! the paper looks like it is pretty old. Hopefully, @SándorKovács could add something useful.
$endgroup$
– schematic_boi
7 hours ago
add a comment |
$begingroup$
great! the paper looks like it is pretty old. Hopefully, @SándorKovács could add something useful.
$endgroup$
– schematic_boi
7 hours ago
$begingroup$
great! the paper looks like it is pretty old. Hopefully, @SándorKovács could add something useful.
$endgroup$
– schematic_boi
7 hours ago
$begingroup$
great! the paper looks like it is pretty old. Hopefully, @SándorKovács could add something useful.
$endgroup$
– schematic_boi
7 hours ago
add a comment |
schematic_boi is a new contributor. Be nice, and check out our Code of Conduct.
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