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

TL;DR

This paper proves the rationality and irreducibility of the moduli space of instanton vector bundles on projective 3-space, extending known results to arbitrary rank and charge, and explores their categorical structure.

Contribution

It establishes the rationality and irreducibility of the moduli space for all ranks and charges, and analyzes the categorical structure of instantons.

Findings

Moduli space is rational and irreducible for all ranks and charges.

Mathematical instantons form a monoidal category.

Detailed analysis of the Barth-Hulek monad-construction supports the results.

Abstract

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.