On rank 3 instanton bundles on $\mathbb{P}^3

We investigate rank $3$ instanton vector bundles on $\mathbb{P}^3$ of charge $n$ and its correspondence with rational curves of degree $n+3$. For $n=2$ we present a correspondence between stable rank $3$ instanton bundles and stable rank $2$ reflexive linear sheaves of Chern classes $(c_1,c_2,c_3)=(-1,3,3)$ and we use this correspondence to compute the dimension of the family of stable rank $3$ instanton bundles of charge $2$. Finally, we use the results above to prove that the moduli space of rank $3$ instanton bundles on $\mathbb{P}^3$ of charge $2$ coincides with the moduli space of rank $3$ stable locally free sheaves on $\mathbb{P}^3$ of Chern classes $(c_1,c_2,c_3)=(0,2,0)$. This moduli space is irreducible, has dimension 16 and its generic point corresponds to a \textcolor{black}{generalized} t`Hooft instanton bundle.