The moduli space of sheaves over a smooth variety is in general not separated. That is, there exists a flat family of coherent sheaves over a punctured disk which extends to a flat family of coherent sheaves over a disk in several ways. Could anyone give me easy examples? 



Non-separatedness of moduli space of sheaves
There is a vector bundle $E$ on $\mathbb{P}^1 \times \mathbb{A}^1$ whose restriction to $\mathbb{P}^1 \times (\mathbb{A}^1 - \{ 0\})$ is isomorphic to $\mathcal{O}^2$, but whose restriction to $\mathbb{P}^1 \times 0$ is $\mathcal{O}(-1)\oplus \mathcal{O}(1)$.


One can take $E$ to be the cokernel of the map
$$ (x, y, t) : \mathcal{O}(-1) \to \mathcal{O}\oplus \mathcal{O} \oplus\mathcal{O}(-1),$$
where $x$ and $y$ are the coordinates on $\mathbb{P}^1$ and $t$ is the coordinate on $\mathbb{A}^1$. Indeed, for nonzero $t$ the map has a section, and for $t=0$ we get $\mathcal{O}(-1)\oplus\mathcal{O}(1)$.


The whole point of course is that $\mathcal{O}(-1)\oplus \mathcal{O}(1)$ is not semistable.


