Torelli Groups and Geometry of Moduli Spaces of Curves

TL;DR

This paper explores the homological and geometric properties of Torelli groups and moduli spaces of curves, applying Johnson's work and Hodge theory to derive new results on Picard groups, normal functions, and monodromy groups.

Contribution

It provides new insights into the structure of moduli spaces, classifies natural normal functions, and generalizes classical conjectures using Johnson's homology results and Hodge modules.

Findings

Picard groups of moduli spaces are finitely generated

Classification of natural normal functions over moduli spaces

Computation of monodromy group of nth roots of the canonical bundle

Abstract

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.