  In this paper we give an exposition of Dennis Johnson's work on the first
homology of the Torelli groups and show how it can be applied, alone and in
concert with Saito's theory of Hodge modules, to study the geometry of moduli
spaces of curves. For example, we show that the picard groups of moduli spaces
of curves with a fixed level structure are finitely generated, classify all
"natural" normal functions defined over moduli spaces of curves with a fixed
level, and also "compute" the height paring between cycles over moduli spaces
of curves which are homologically trivial and disjoint over the generic point.
Several new sections have been added. These apply the results on normal
functions to prove generalizations of the classical Franchetta conjecture for
curves and abelian varieties. In one section, the monodromy group of nth roots
of the canonical bundle is computed.
