New divisors in the boundary of the instanton moduli space

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.

arXiv comment30 pages