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.
We study the moduli space $I_{n,r}$ of rank-$2r$ symplectic instanton vector bundles on $mathbb{P}^3$ with $rge2$ and second Chern class $nge r+1, n-requiv 1(mathrm{mod}2)$. We introduce the notion of tame symplectic instantons by excluding a kind of
pathological monads and show that the locus $I^*_{n,r}$ of tame symplectic instantons is irreducible and has the expected dimension, equal to $4n(r+1)-r(2r+1)$. The proof is inherently based on a relation between the spaces $I^*_{n,r}$ and the moduli spaces of t Hooft instantons
We construct a compactification $M^{mu ss}$ of the Uhlenbeck-Donaldson type for the moduli space of slope stable framed bundles. This is a kind of a moduli space of slope semistable framed sheaves. We show that there exists a projective morphism $gam
ma colon M^{ss} to M^{mu ss}$, where $M^{ss}$ is the moduli space of S-equivalence classes of Gieseker-semistable framed sheaves. The space $M^{mu ss}$ has a natural set-theoretic stratification which allows one, via a Hitchin-Kobayashi correspondence, to compare it with the moduli spaces of framed ideal instantons.
