I am given to understand that the moduli space $M_k^G$ of $G$ vector bundles with second Chern class $c_2=k$ over an algebraic curve/variety (for me a Riemann surface is enough/projective space for varieties) is a non-compact space. I am familiar with the Uhlenbeck compactification
$$
\widetilde{M}_k^G = M_{k}^G \cup (M_{k-1}^G \times \mathbb{R}^4) \cup (M_{k-2}^G \times \mathrm{Sym}^2\mathbb{R}^4) \cup \ldots \cup \mathrm{Sym}^k\mathbb{R}^4
$$
Apparently, there is another kind of compactification using **sheaves**. Vector bundles are locally free sheaves so I am trying to understand, in other words, the moduli space of locally free sheaves. Now, if we include more general sheaves, not necessarily locally free, we get the moduli space of coherent (torsion free) sheaves.

**Is this statement correct? What is the precise idea behind this compactification, what is its relation to the Uhlenbeck one and is it true that this compact moduli space is now singular? And what if we have a smooth surface, e.g. $\mathbb{P}^n$ or any of the Hirzebruch surfaces?**

I would appreciate very much any reference on this "sheaf" compactification.

P.S. I think this is referred to as **Gieseker compactification of the moduli space of sheaves** but a Google search wont give me much info and Huybrecht's book is way to hard and lengthy.