On extensions of the Torelli map

TL;DR

This paper characterizes the divisors on the moduli space of curves that come from extensions of the Torelli map to toroidal compactifications, revealing a specific extremal face of the nef cone.

Contribution

It explicitly describes the 2-dimensional extremal face of the nef cone of ar{M}_g associated with these divisors, extending understanding of the Torelli map's geometric properties.

Findings

Identifies a 2-dimensional extremal face of the nef cone of ar{M}_g.

Describes the divisors arising from extensions of the Torelli map.

Provides an explicit description of this extremal face.

Abstract

The divisors on $\bar{\operatorname{M}}_g$ that arise as the pullbacks of ample divisors along any extension of the Torelli map to any toroidal compactification of $\operatorname{A}_g$ form a 2-dimensional extremal face of the nef cone of $\bar{\operatorname{M}}_g$, which is explicitly described.