Rationality and categorical properties of the moduli of instanton bundles on the projective 3-space

We prove the rationality and irreducibility of the moduli space of mathematical instanton vector bundles of arbitrary rank and charge on ${\mathbb{P}}^3$. In particular, the result applies to the rank-2 case. This problem was first studied by Barth, Ellingsrud-Stromme, Hartshorne, Hirschowitz-Narasimhan in the late 1970s. We also show that the mathematical instantons of variable rank and charge form a monoidal category. The proof is based on an in-depth analysis of the Barth-Hulek monad-construction and on a detailed description of the moduli space of (framed and unframed) stable bundles on Hirzebruch surfaces.