TITLE: Simple sheaves are smooth points in the Quot scheme
QUESTION [7 upvotes]: Let we consider the Quot scheme $Q(X,N,P)$ of coherent sheaves on an algebraic variety with given Hilbert polinomial $P$ and quotient of $\mathcal{O}_X^N$, i.e. all $\mathcal{F}$ of the form $\mathcal{O}_X^N\longrightarrow\mathcal{F}\longrightarrow 0$. 
Recall that a simple sheaves $\mathcal{F}$ is a coherent sheaves such that $End(\mathcal{F})$ is trivial. Then, why is the Quot scheme smooth in the point represented by $\mathcal{F}$, where $\mathcal{F}$ is simple? 
I know that the tangent space of the Quot scheme in the point
$$q=[\mathcal{O}_X^N\stackrel{q}{\longrightarrow}\mathcal{F}\longrightarrow 0]$$
is $$T_qQuot\cong Hom(\mathcal{G},\mathcal{F}),$$ where $\mathcal{G}:=\ker q$, but I am not able to answer the above question only from this.

REPLY [2 votes]: The claim is not true in general! The obstructions for a coherent (semistable) sheaves to be a smooth point in the Quot scheme are contained in the second Ext group $Ext^2(\mathcal{F},\mathcal{F})$. Even if $\mathcal{F}$ is simple, this group can be very big and there is no matter because the obstructions must vanish.
Anyway if $X$ is a surface (with trivial canonical bundle) we can use Serre Duality to claim that $Ext^2(\mathcal{F},\mathcal{F})\cong Hom(\mathcal{F},\mathcal{F})$ that is trivial, by the assumption that $\mathcal{F}$ is simple. 
References for this can be: 1) Huybrechts, Lehn, "The geometry of moduli spaces of sheaves" section 4.5. 2) Mukai, "Moduli of vector bundles on K3 surfaces and symplectic manifolds" Sugaku Expositions, vol.1, n.2, 1988, pag. 151-153.