Moduli space of vector bundles open in moduli of sheaves












4














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!










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  • 2




    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.
    – Sasha
    Nov 27 '18 at 22:15
















4














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!










share|cite|improve this question




















  • 2




    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.
    – Sasha
    Nov 27 '18 at 22:15














4












4








4


1





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!










share|cite|improve this question















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






share|cite|improve this question















share|cite|improve this question













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share|cite|improve this question








edited Nov 27 '18 at 19:35

























asked Nov 27 '18 at 18:56









Rust Q

1278




1278








  • 2




    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.
    – Sasha
    Nov 27 '18 at 22:15














  • 2




    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.
    – Sasha
    Nov 27 '18 at 22:15








2




2




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.
– Sasha
Nov 27 '18 at 22:15




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.
– Sasha
Nov 27 '18 at 22:15















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