Extending Torelli map to toroidal compactifications of Siegel space

TL;DR

This paper proves the Torelli map extends regularly to the perfect cone compactification and shares a neighborhood with the Voronoi compactification, but not to the Igusa monoidal transform for genus ≥9, disproving a long-standing conjecture.

Contribution

It establishes the regularity of the Torelli map extension to the perfect cone compactification and compares it with Voronoi and Igusa compactifications, disproving a 1973 conjecture.

Findings

Torelli map extends regularly to the perfect cone compactification.

Voronoi and perfect cone compactifications share a Zariski open neighborhood.

The map to the Igusa monoidal transform is not regular for genus ≥9.

Abstract

It has been known since the 1970s that the Torelli map $M_g \to A_g$, associating to a smooth curve its jacobian, extends to a regular map from the Deligne-Mumford compactification $\bar{M}_g$ to the 2nd Voronoi compactification $\bar{A}_g^{vor}$. We prove that the extended Torelli map to the perfect cone (1st Voronoi) compactification $\bar{A}_g^{perf}$ is also regular, and moreover $\bar{A}_g^{vor}$ and $\bar{A}_g^{perf}$ share a common Zariski open neighborhood of the image of $\bar{M}_g$. We also show that the map to the Igusa monoidal transform (central cone compactification) is NOT regular for $g\ge9$; this disproves a 1973 conjecture of Namikawa.