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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?

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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.

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