A rank \(2\) instanton sheaf on the complex projective space \({\mathbb P} ^3\) is a torsion-free rank \(2\) sheaf \(E\) with \(\det E\simeq {\mathcal O}_{{\mathbb P}^3}\) and such that \[H^0(E(-1))=H^1(E(-2))=H^2(E(-2))=H^3(E(-3))=0.\] Such sheaves have only one non-trivial Chern class \(c_2(E)\), which can be identified with an integer via the canonical isomorphism \(H^4({\mathbb P}^3, {\mathbb Z})\simeq {\mathbb Z}\).
One can consider the moduli space \({\mathcal I} (n)\) of rank \(2\) locally free instanton sheaves \(E\) with \(c_2(E)=n\) and the moduli space \({\mathcal L} (n)\) of rank \(2\) instanton sheaves \(E\) with \(c_2(E)=n\). Whereas \({\mathcal I} (n)\) is a smooth affine irreducible variety, \({\mathcal L} (n)\) can be singular with several irreducible components.
Every instanton sheaf corresponds to a representation of some special quiver with relations, consisting of \(3\) vertices and \(4\) arrows between two pairs of vertices. The main aim of the paper is to study compactifications of \({\mathcal I} (n)\) and \({\mathcal L} (n)\) within King’s moduli spaces of \(\theta\)-semistable representations of this quiver. The main theorem achieves this aim for \(n=1\), in which case the authors obtain a complete description of such compactifications.