Assume $X$ is a smooth projective scheme over some algebraically closed field $k$. Let $M_1$ and $M_2$ be two moduli spaces of vector bundles on $X$.

The first space contains just one point, this point is given by a bundle $F$ with $dim(Ext^1(F,F))=0$ and $dim(Ext^2(F,F))=1$. So we have $M_1=Spec(k)$.

The second space also contains just one point. This point is given by a bundle $E$, such that $dim(Ext^1(E,E))=dim(Ext^2(E,E))=1$.

How can i determine the structure of $M_2$?

Since $dim(M_2)=0 < dim(Ext^1(E,E)=1$ i think $M_2$ cannot be smooth. And since $Ext^2(E,E)=1$ the bundle is possibly obstructed.

I think $M_2$ is possibly a non reduced point $Spec(k[x]/(x^n))$ for some $n\in \mathbb{N}$.

Is this idea corerct? How to find the number $n$? Is there a general way to determine the local structure of a moduli space at some point by just knowing the $Ext$-groups of that point? What are other possible structures for $M_2$?