Is there a standard reference for the fact that, in an appropriate algebraic-geometrical context, the tangent space at the point $[E]$ to the moduli space $\mathcal M$ is something like $\operatorname{Ext}^1(E, E)$? I am primary interested in two situations:

- $\mathcal M$ is the moduli space of semi-stable coherent torsion-free sheaves on $\mathbb CP^2$ with given Hilbert polynomial,
- or $\mathcal M$ is the moduli space of semi-stable finite-dimensional complex representations of finite acyclic quiver with given dimension vector,

and in both situation, if the coefficients of the Hilbert polynomial/dimension vector are coprime, there is a smooth projective variety that is a fine moduli space $\mathcal M$.

Unfortunately, I do not know any deformation theory, so a study book on it that also proves something like $T_{[E]} \mathcal M=\operatorname{Ext}^1(E, E)$ would be welcome.

The Geometry of moduli space of sheavesby Huybrechts and Lehn -- see e.g. Corollary 4.5.2. – abx Oct 31 '16 at 20:44