Normal Functions and the Geometry of Moduli Spaces of Curves

TL;DR

This paper employs normal functions to address geometric questions about moduli spaces of curves, including computing cohomology classes and exploring slope inequalities, with implications for algebraic geometry.

Contribution

It introduces the use of normal functions to solve problems related to moduli spaces of curves and refines understanding of slope inequalities and height jumping.

Findings

Computed the cohomology class of a specific pullback in moduli space.

Provided new insights into slope inequalities for curves.

Discussed the role of height jumping in geometric inequalities.

Abstract

In this paper normal functions (in the sense of Griffiths) are used to solve and refine geometric questions about moduli spaces of curves. The first application is to a problem posed by Eliashberg: compute the class in the cohomology of M_{g,n}^c of the pullback of the zero section of the universal jacobian along the section that takes [C;x_1,...,x_n] to Sum d_j x_j in Jac (C), where d_1 + ... + d_n = 0. The second application is to slope inequalities of the type discovered by Moriwaki. There is also a discussion of height jumping and its relevance to slope inequalilties.