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.

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