Let $X$ be a surface, and $i:C\subset X$ be a smooth curve. Let $A$ be a line bundle on $C$, and $E$ be a vector bundle of rank $r$ on $X$.
Suppose there is a surjection: $E\longrightarrow i_*A\longrightarrow 0\\$, then we get an exact sequence $\qquad\qquad\qquad\qquad\qquad0\longrightarrow F\longrightarrow E\longrightarrow i_*A\longrightarrow 0$.
Now $i_*A$ is torsion, we have $(i_*A)^*=0$. Therefore taking the dual sequence, we get
$\qquad\qquad\qquad\qquad\qquad 0\longrightarrow E^*\longrightarrow F^*\longrightarrow Q\longrightarrow 0$.
What is the co-kernel $Q$? How do we get it? Thanks in advance!