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.




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



Families of torsion-free sheaves whose length jumps
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.


