Admissible Pairs vs Gieseker--Maruyama

A morphism of the moduli functor of admissible semistable pairs to the Gieseker – Maruyama moduli functor (of semistable coherent torsion-free sheaves) with the same Hilbert polynomial on the surface, is constructed. It is shown that these functors are isomorphic, and the moduli scheme for semistable admissible pairs $((\tilde{S},\tilde{L}),\tilde{E})$ is isomorphic to the Gieseker – Maruyama moduli scheme. The considerations involve all components of moduli functors and corresponding moduli scheme as they exist.