complex geometry - question about torsion sheaf

Let's assume for simplicity that $M$ is a smooth, complex, projective variety.

The set of points where the coherent subsheaf $mathcal{F}$ is not locally free is a proper closed subset of $M$ (Hartshorne, Algebraic Geometry, Chapter II, ex. 5.8), so the stalk of $ker(det(j))$ at the generic point is zero, i.e. it is a torsion sheaf.

Moreover, you can say more. Indeed, since $mathcal{E}$ is locally free and $mathcal{E} /mathcal{F}$ is torsion-free, it follows that $mathcal{F}$ is a reflexive sheaf (Hartshorne, Stable Reflexive Sheaves, Theorem 1.1), so it is locally free except along a closed subset of codimension $geq 3$ (same reference, Corollary 1.4).

In particular, if $M$ is a curve or a surface then $ker(det(j))$ is zero.

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