Moduli space of vector bundles open in moduli of sheaves
$begingroup$
Let $X$ be a smooth projective variety with polarization $mathcal{O}_X(1)$. For fixed Chern class $r,c_1,c_2....$, do we have an OPEN embedding
$$M^{ss}(X,r,c_1...)_{vec}subset M^{ss}(X,r,c_1...)$$
of moduli space (Gieseker)-semistable vector bundles into the moduli space of semistable sheaves with the above Chern classes? ( I guess I should assume there is such a vector bundle to start with)
Also, since $mu$-stable implies stable, is the moduli space of $mu$-stable vector bundles with the above Chern character openly embedded in $M^{ss}(X,r,c_1...)_{vec}$?
Thanks!
algebraic-geometry moduli-space
asked Nov 27 '18 at 18:56
Rust QRust Q
1328
$endgroup$
add a comment |
$begingroup$
Let $X$ be a smooth projective variety with polarization $mathcal{O}_X(1)$. For fixed Chern class $r,c_1,c_2....$, do we have an OPEN embedding
$$M^{ss}(X,r,c_1...)_{vec}subset M^{ss}(X,r,c_1...)$$
of moduli space (Gieseker)-semistable vector bundles into the moduli space of semistable sheaves with the above Chern classes? ( I guess I should assume there is such a vector bundle to start with)
Also, since $mu$-stable implies stable, is the moduli space of $mu$-stable vector bundles with the above Chern character openly embedded in $M^{ss}(X,r,c_1...)_{vec}$?
Thanks!
algebraic-geometry moduli-space
asked Nov 27 '18 at 18:56
Rust QRust Q
1328
$endgroup$
2
$begingroup$
I am not sure that the property of being locally free is preserved by S-equivalence, and so that the locus of vector bundles is well-defined for the moduli of semistable sheaves. On a contrary, the locus of STABLE vector bundles is a well-defined open subscheme in the stable locus of the moduli space.
$endgroup$
– Sasha
Nov 27 '18 at 22:15
add a comment |
$begingroup$
Let $X$ be a smooth projective variety with polarization $mathcal{O}_X(1)$. For fixed Chern class $r,c_1,c_2....$, do we have an OPEN embedding
$$M^{ss}(X,r,c_1...)_{vec}subset M^{ss}(X,r,c_1...)$$
of moduli space (Gieseker)-semistable vector bundles into the moduli space of semistable sheaves with the above Chern classes? ( I guess I should assume there is such a vector bundle to start with)
Also, since $mu$-stable implies stable, is the moduli space of $mu$-stable vector bundles with the above Chern character openly embedded in $M^{ss}(X,r,c_1...)_{vec}$?
Thanks!
algebraic-geometry moduli-space
asked Nov 27 '18 at 18:56
Rust QRust Q
1328
$endgroup$
Let $X$ be a smooth projective variety with polarization $mathcal{O}_X(1)$. For fixed Chern class $r,c_1,c_2....$, do we have an OPEN embedding
$$M^{ss}(X,r,c_1...)_{vec}subset M^{ss}(X,r,c_1...)$$
of moduli space (Gieseker)-semistable vector bundles into the moduli space of semistable sheaves with the above Chern classes? ( I guess I should assume there is such a vector bundle to start with)
Also, since $mu$-stable implies stable, is the moduli space of $mu$-stable vector bundles with the above Chern character openly embedded in $M^{ss}(X,r,c_1...)_{vec}$?
Thanks!
algebraic-geometry moduli-space
algebraic-geometry moduli-space
asked Nov 27 '18 at 18:56
Rust QRust Q
1328
asked Nov 27 '18 at 18:56
Rust QRust Q
1328
edited Nov 27 '18 at 19:35
Rust Q
asked Nov 27 '18 at 18:56
Rust QRust Q
1328
asked Nov 27 '18 at 18:56
Rust QRust Q
1328
asked Nov 27 '18 at 18:56
Rust QRust Q
1328
1328
2
$begingroup$
I am not sure that the property of being locally free is preserved by S-equivalence, and so that the locus of vector bundles is well-defined for the moduli of semistable sheaves. On a contrary, the locus of STABLE vector bundles is a well-defined open subscheme in the stable locus of the moduli space.
$endgroup$
– Sasha
Nov 27 '18 at 22:15
add a comment |
2
$begingroup$
I am not sure that the property of being locally free is preserved by S-equivalence, and so that the locus of vector bundles is well-defined for the moduli of semistable sheaves. On a contrary, the locus of STABLE vector bundles is a well-defined open subscheme in the stable locus of the moduli space.
$endgroup$
– Sasha
Nov 27 '18 at 22:15
2
2
$begingroup$
I am not sure that the property of being locally free is preserved by S-equivalence, and so that the locus of vector bundles is well-defined for the moduli of semistable sheaves. On a contrary, the locus of STABLE vector bundles is a well-defined open subscheme in the stable locus of the moduli space.
$endgroup$
– Sasha
Nov 27 '18 at 22:15
$begingroup$
I am not sure that the property of being locally free is preserved by S-equivalence, and so that the locus of vector bundles is well-defined for the moduli of semistable sheaves. On a contrary, the locus of STABLE vector bundles is a well-defined open subscheme in the stable locus of the moduli space.
$endgroup$
– Sasha
Nov 27 '18 at 22:15
add a comment |
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$begingroup$
I am not sure that the property of being locally free is preserved by S-equivalence, and so that the locus of vector bundles is well-defined for the moduli of semistable sheaves. On a contrary, the locus of STABLE vector bundles is a well-defined open subscheme in the stable locus of the moduli space.
$endgroup$
– Sasha
Nov 27 '18 at 22:15