  In an earlier paper [Acta Mathematica, v. 176, 1996, 145-169,
alg-geom/9505024 ] the present authors and Dennis Sullivan constructed the
universal direct system of the classical Teichm\"uller spaces of Riemann
surfaces of varying genus. The direct limit, which we called the universal
commensurability Teichm\"uller space, $T_{infty}$, was shown to carry on it a
natural action of the universal commensurability mapping class group,
$MC_{\infty}$. In this paper we identify an interesting cofinal sub-system
corresponding to the tower of finite-sheeted characteristic coverings over any
fixed base surface. Utilizing a certain subgroup inside $MC_{\infty}$,
(associated intimately to the inverse system of characteristic coverings), we
can now descend to an inductive system of moduli spaces, and construct the
direct limit ind-variety $M_{\infty}$.
  Invoking curvature properties of Quillen metrics on determinant bundles, and
naturality under finite coverings of Weil-Petersson forms, we are able to
construct on $M_{\infty}$ the natural sequence of determinant of cohomology
line bundles, as well as the Mumford isomorphisms connecting these.
