New divisors in the boundary of the instanton moduli space

TL;DR

This paper investigates the boundary structure of the instanton moduli space on projective 3-space, constructing new divisors that correspond to singular sheaves with rational curve singularities, enriching the understanding of its compactification.

Contribution

It introduces new irreducible components of the boundary of the instanton moduli space, characterized by rank 2 torsion free sheaves with rational curve singularities.

Findings

Constructed irreducible components of the boundary of the moduli space.

Boundary components have dimension 8n-4 and lie in the smooth locus.

Boundary sheaves have singularities along rational curves.

Abstract

Let ${\mathcal I}(n)$ denote the moduli space of rank $2$ instanton bundles of charge $n$ on ${\mathbb P}^3$. We know from several authors that ${\mathcal I}(n)$ is an irreducible, nonsingular and affine variety of dimension $8n-3$. Since every rank $2$ instanton bundle on ${\mathbb P}^3$ is stable, we may regard ${\mathcal I}(n)$ as an open subset of the projective Gieseker--Maruyama moduli scheme ${\mathcal M}(n)$ of rank $2$ semistable torsion free sheaves $F$ on ${\mathbb P}^3$ with Chern classes $c_1=c_3=0$ and $c_2=n$, and consider the closure $\overline{{\mathcal I}(n)}$ of ${\mathcal I}(n)$ in ${\mathcal M}(n)$. We construct some of the irreducible components of dimension $8n-4$ of the boundary $\partial{\mathcal I}(n):=\overline{{\mathcal I}(n)}\setminus{\mathcal I}(n)$. These components generically lie in the smooth locus of ${\mathcal M}(n)$ and consist of rank $2$ torsion free instanton sheaves with singularities along rational curves.