Let $X$ be a smooth projective surface over the field of complex numbers. Suppose I have an injective morphism of invertible sheaves on $X$, say,
$$0\rightarrow M\rightarrow N.$$
Consider the cokernel $T$, a torsion sheaf. Assume the support of $T$ is a smooth curve $C$ on $X$. What can we say about $T|_C$? Will it be torsion-free and hence locally free? Or can it have some torsion.
If you restrict the exact sequence $$ 0 \to M \to N \to T \to 0 $$ to $C$, you will get a right exact sequence $$ M\vert_C \to N\vert_C \to T\vert_C \to 0. $$ By definition of $C$ the first map vanishes, hence $T\vert_C \cong N\vert_C$ is a line bundle on $C$.