The Torelli group Tg is the group of isotopy classes of diffeomorphisms of a compact orientable surface of genus g that act trivially on the homology of the surface. The aim of this paper is to show how facts about the homology of the Torelli group imply interesting results about algebraic curves. We begin with an exposition of some of Dennis Johnson's work on the Torelli groups. We then show how these results imply that the Picard group of the moduli space of curves of genus g ≥ 3 with a level-J structure is finitely generated. A classification of all “natural” normal functions over the moduli space of curves of genus g ≥ 3 and a level / structure is obtained by combining Johnson's results with M. Saito's theory of Hodge modules. This is used to prove results that generalize the classical Franchetta Conjecture to the generic curve of genus g with n marked points and a level-/ structure. Other applications are given, for example, to computing heights of cycles defined over a moduli space of curves.

1. Introduction

The Torelli group Tg is the kernel of the natural homomorphism Γg → Spp(ℤ) from the mapping class group in genus g to the group of 2g x 2g integral symplectic matrices. It accounts for the difference between the topology of Ag, the moduli space of principally polarized abelian varieties of dimension g, and Mg, the moduli space of smooth projective curves of genus g, and therefore should account for some of the difference between their geometries.