Torelli Groups and Geometry of Moduli Spaces of Curves

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.

38 pages, AMS-LaTeX. Abstract of revised version: This is a significanly revised and expanded version of the original paper. Several significant typos in the section on heights have been corrected and four new sections have been added. Among the new results are proofs of several generalizations of the classical Franchetta Conjecture to curves with a level structure and n marked points. Another section contains a computation of the monodromy group of nth roots of the canonical bundle. The two

References in corpus (1)

• Completions of mapping class groups and the cycle $C-C^{-}$

Cited by in corpus (21)

• Representation theory and homological stability
• Normal Functions and the Geometry of Moduli Spaces of Curves
• Abelian quotients of subgroups of the mapping class group and higher Prym representations
• The Torelli group and congruence subgroups of the mapping class group
• Geometric Proofs of Some Results of Morita
• Irreducible Sp-representations and subgroup distortion in the mapping class group
• The second rational homology group of the moduli space of curves with level structures
• The Johnson homomorphism and its kernel
• On power subgroups of mapping class groups
• Mapping Class Groups and Moduli Spaces of Curves
• The abelianization of the level L mapping class group
• Normal functions for algebraically trivial cycles are algebraic for arithmetic reasons
• Torelli buildings and their automorphisms
• Abelian covers of surfaces and the homology of the level L mapping class group
• Relative Pro-$\ell$ Completions of Mapping Class Groups
• The Hodge de Rham theory of relative Malcev completion
• Normal Functions over Locally Symmetric Varieties
• On the integral Hodge conjecture for real abelian threefolds
• Infinitesimal presentations of the Torelli groups
• Refinements of Katz-Sarnak theory for the number of points on curves over finite fields
• Galois actions on fundamental groups of curves and the cycle $C-C^{-}$