Singular locus of instanton sheaves on $\mathbb{P}^3$

We prove that the singular locus of a rank 2 instanton sheaf $E$ on $\mathbb{P}^3$ which is not locally free has pure dimension 1. Moreover, we also show that the dual and double dual of $E$ are isomorphic locally free instanton sheaves, and that the sheaves $\mathcal{E}xt^1(E,\mathcal{O}_{\mathbb{P}^3)$ and $E^{\vee\vee}/E$ are rank $0$ instantons. We also provide explicit examples of instanton sheaves of rank $3$ and $4$ illustrating that all of these claims are false for higher rank instanton sheaves.

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