On a new compactification of moduli of vector bundles on a surface, III: Functorial approach

A new compactification for the scheme of moduli for Gieseker-stable vector bundles with prescribed Hilbert polynomial, on the smooth projective polarized surface (S,L), is constructed. Families of locally free sheaves on the surface S are completed by locally free sheaves on schemes which are modifications of S. Gieseker -- Maruyama moduli space has a birational morphism onto the new moduli space. We propose the functor for families of pairs polarized scheme -- vector bundle with moduli space of such type.