Let $X$ be a normal Gorenstein complex surface and $F$ be a rank one reflexive sheaf on $X$. I'm trying to find some ways to determine local Ext $\mathcal{E}xt^i_X(F,F)$.

For $i=0$, by the normality of $F$, I think we have $\mathcal{H}om_X(F,F)=\mathcal{O}_X$. Is there any similar result for $i>0$ (e.g. $\mathcal{E}xt^i_X(F,F)=0$ for $i>0$)?

The only thing now I can show is that $\mathcal{E}xt^i_X(F,F)$ is supported on points for $i>0$.