  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.

Algebraic Geometry;Algebraic Geometry