# Families of torsion-free sheaves whose length jumps

For a long time, I had a false belief that the space/stack $$\text{Coh}^{tf}_{c_1,c_2}S$$ of torsion-free sheaves $$\mathcal{E}$$ on a smooth algebraic surface $$S$$ was not connected, since if you take its map into its double dual $$0\ \to \ \mathcal{E}\ \to\ \mathcal{E}^{\vee\vee}\ \to \ \mathcal{Q}\ \to\ 0$$ the length of the torsion sheaf $$\mathcal{Q}$$ gives another invariant labelling connected components.

What are some "typical" examples of (say, flat one parameter) families of torsion-free sheaves where the torsion quotient jumps in length?

Consider the sheaf $$F$$ on $$\mathbb{P}^2 \times \mathbb{A}^1$$ defined from the exact sequence $$0 \to F \longrightarrow \mathcal{O}(-1)^{\oplus 3} \stackrel{(x,y,tz)}\longrightarrow \mathcal{O} \longrightarrow \mathcal{O}_P \to 0,$$ where $$(x,y,z)$$ are the homogeneous coordinates on $$\mathbb{P}^2$$, $$t$$ is the coordinate on $$\mathbb{A}^1$$, and $$P$$ is the point $$((0,0,1),0)$$.
On the one hand, if one restricts to the fiber over $$0 \ne t \in \mathbb{A}^1$$, one obtains $$F_t \cong \Omega_{\mathbb{P}^2},$$ in particular, this sheaf is locally free.
On the other hand, over $$0 \in \mathbb{A}^1$$, one obtains an exact sequence $$0 \to F_0 \to \mathcal{O}(-2) \oplus \mathcal{O}(-1) \to \mathcal{O}_P \to 0,$$ hence this sheaf is not locally free.