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. We work over the field $k=\bar k$ of characteristic zero. Families of locally free sheaves on the surface $S$ are completed with locally free sheaves on schemes which are modifications of $S$. The 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.
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