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 $\overline{ M_g}$ of Simpson's $p$-semistable coherent sheaves and a canonical dominating morphism from the universal Hilbert scheme over $\overline{ M_g}$ to a compactified universal Picard.
Categories: alg-geom math.AG