Isomorphism of compactifications of vector bundles moduli: nonreduced moduli

We continue the study of the compactification of the moduli scheme for Gieseker-semistable vector bundles on a nonsingular irreducible projective algebraic surface $S$ with polarization $L$, by locally free sheaves. The relation of main components of the moduli functor for admissible semistable pairs and main components of the Gieseker–Maruyama moduli functor (for semistable torsion-free coherent sheaves) with the same Hilbert polynomial on the surface $S$ is investigated.
The compactification of interest arises when families of Gieseker-semistable vector bundles $E$ on the nonsingular polarized projective surface $(S, L)$ are completed by vector bundles $\widetilde E$ on projective polarized schemes $(\widetilde S, \widetilde L)$ of special form. The form of the scheme $\widetilde S$, of its polarization $\widetilde L$ and of the vector bundle $\widetilde E$ is described in the text. The collection $((\widetilde S, \widetilde L), \widetilde E)$ is called a semistable admissible pair. Vector bundles $E$ on the surface $(S, L)$ and $\widetilde E$ on schemes $(\widetilde S, \widetilde L)$ are supposed to have equal ranks and Hilbert polynomials which are compute with respect to polarizations $L$ and $\widetilde L$, respectively. Pairs of the form $((S, L), E)$ named as $S$-pairs are also included into the class under the scope. Since the purpose is to study the compactification of moduli space for vector bundles, only families which contain $S$-pairs are considered.
We build up the natural transformation of the moduli functor for admissible semistable pairs to the Gieseker–Maruyama moduli functor for semistable torsion-free coherent sheaves on the surface $(S,L)$, with same rank and Hilbert polynomial. It is demonstrated that this natural transformation is inverse to the natural transformation built in the preceding paper and defined by the standard resolution of a family of torsion-free coherent sheaves with a possibly nonreduced base scheme. The functorial isomorphism constructed determines the scheme isomorphism of compactifications of moduli space for semistable vector bundles on the surface $(S,L)$.