This is my question:

Given a noetherian scheme $X$, the structural sheaf $\mathcal{O}_X$ is a coherent sheaf, so every locally free sheaf is coherent. This means that the family of stable locally free sheaves (with fixed Hilbert polynomial) is parametrized by a subset $S$ of the moduli space $M$ of stable sheaves on $X$. Is it an open subset? i.e. is the locally free condition an open condition?

I use the definition that a stable sheaves is a pure sheaves with strictly decreasing reduced Hilbert polynomial on proper subsheaves, so ,in particular, every sheaf in $M$ is torsion free. But torsion free sheaves are not locally free in general, so $S$ is a proper subset.

I'm wondering also if there exists an example of a torsion free sheaf and a locally free sheaf with the same Mukai vector. If this is not the case, I can obtain the required subset by the union of the connected components, relative to Mukai vectors corresponding to locally free sheaves.

Thank you!

This is my question:

Given a projective noetherian scheme $X$, the structural sheaf $\mathcal{O}_X$ is a coherent sheaf, so every locally free sheaf is coherent. This means that the family of stable locally free sheaves (with fixed Hilbert polynomial) is parametrized by a subset $S$ of the moduli space $M$ of stable sheaves on $X$. Is it an open subset? i.e. is the locally free condition an open condition?

I use the definition that a stable sheaves is a pure sheaves with strictly decreasing reduced Hilbert polynomial on proper subsheaves, so ,in particular, every sheaf in $M$ is torsion free. But torsion free sheaves are not locally free in general, so $S$ is a proper subset.

I'm wondering also if there exists an example of a torsion free sheaf and a locally free sheaf with the same Mukai vector. If this is not the case, I can obtain the required subset by the union of the connected components, relative to Mukai vectors corresponding to locally free sheaves.

Thank you!

P.S.: In general which are the minimal conditions on $ X $ to have that the locally free condition is an open condition?

This is my question:

Given a noetherian scheme $X$, the structural sheaf $\mathcal{O}_X$ is a coherent sheaf, so every locally free sheaf is coherent. This means that the family of stable locally free sheaves (with fixed Hilbert polinomial) is parametrized by a subset $S$ of the moduli space $M$ of stable sheaves on $X$. Is it an open subset? i.e. is the locally free condition an open condition?

I use the definition that a stable sheaves is a pure sheaves with strictly decreasing reduced Hilbert polinomial on proper subsheaves, so ,in particular, every sheaf in $M$ is torsion free. But torsion free sheaves are not locally free in general, so $S$ is a proper subset.

I'm wondering also if there exists an example of a torsion free sheaf and a locally free sheaf with the same Mukai vector. If this is not the case, I can obtain the required subset by the union of the connected components, relative to Mukai vectors corresponding to locally free sheaves.

Thank you!

This is my question:

Given a noetherian scheme $X$, the structural sheaf $\mathcal{O}_X$ is a coherent sheaf, so every locally free sheaf is coherent. This means that the family of stable locally free sheaves (with fixed Hilbert polynomial) is parametrized by a subset $S$ of the moduli space $M$ of stable sheaves on $X$. Is it an open subset? i.e. is the locally free condition an open condition?

I use the definition that a stable sheaves is a pure sheaves with strictly decreasing reduced Hilbert polynomial on proper subsheaves, so ,in particular, every sheaf in $M$ is torsion free. But torsion free sheaves are not locally free in general, so $S$ is a proper subset.

I'm wondering also if there exists an example of a torsion free sheaf and a locally free sheaf with the same Mukai vector. If this is not the case, I can obtain the required subset by the union of the connected components, relative to Mukai vectors corresponding to locally free sheaves.

Thank you!