Let $X$ be the stack of rank $1$ degree $b$ coherent sheaves $E$ with torsion of length at most 1 on an elliptic curve $C$. Let $Y$ be the stack of pairs $E^{'} \subset E$ such that $E \in X$ and $E/E^{'}\cong \mathcal O_x$ for a point $x \in C$. Let $X_1$ be the stack of rank $1$ degree $b$ coherent sheaves whose torsion has length exactly $1$. Question: is it true (and if so, how to see) that $Y$ is the blow-up of $X \times C$ along $X_1$, where $X_1 \subset X \times C$ by $E \mapsto (E, \rm{Torsion}(E))$?