RELATIVE GEOMETRIC INVARIANT THEORY AND UNIVERSAL MODULI SPACES

We expose in detail the principle that the relative geometric invariant theory of equivariant morphisms is related to the GIT for linearizations near the boundary of the G-effective ample cone. We then apply this principle to construct and reconstruct various universal moduli spaces. In particular, we constructed the universal moduli space over of Simpson’s p-semistable coherent sheaves and a canonical rational morphism from the universal Hilbert scheme over to a compactified universal Picard.