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? 



Open subset of the moduli space of stable sheaves on a noetherian scheme
I will answer the second question.


Let $X$ be any smooth projective surface and $P \in X$ a point. Let $F = O\_X \oplus I\_P$. Then $F$ is torsion free but not locally free. But its general deformation is locally free. The reason is that $\dim Ext^1(F,F) = 3$, so a generic deformation of $F$ should erase the singularity of $F$ (of course to turn it into a rigorous proof you should work a bit). 


In case $X = \mathbb{P}^2$ the generic deformation of $F$ can be written as the kernel of a generic map $O\_X \oplus O\_X(1) \oplus O\_X(1) \to O\_X(2)$. 


