On an isomorphism of compactifications of moduli scheme of vector bundles

A morphism of the reduced Gieseker – Maruyama moduli functor (of semistable coherent torsion-free sheaves) on the surface to the reduced moduli functor of admissible semistable pairs with the same Hilbert polynomial, is constructed. It is shown that main components of reduced moduli scheme for semistable admissible pairs $((\widetilde S, \widetilde L), \widetilde E)$ are isomorphic to main components of the reduced Gieseker – Maruyama moduli scheme.