Torsion-free instanton sheaves on the blow-up of P3 at a point

We study an extension of the instanton bundle's definition given by Casnati, Coskun, Genk, and Malaspina for Fano threefolds in order to include non locally-free ones on the blow-up $\widetilde{P^3}$, of the projective 3-space at a point. Using the suggested definition, we show that any reflexive instanton sheaf must be locally free, and that the strictly torsion-free instanton sheaves have singularities of pure dimension 1. We construct examples and study their μ-stability. Additionally, we find that these sheaves contribute to the (partial) compactification of the 't Hooft component of the moduli space of instantons, on $\widetilde{P^3}$. In particular, our non locally-free examples are shown to be smooth and smoothable.