Extending the Torelli map to toroidal compactifications of Siegel space

摘要

It has been known since the 1970s that the Torelli map M _g → A _g, associating to a smooth curve its Jacobian, extends to a regular map from the Deligne-Mumford compactification M? _g to the 2nd Voronoi compactification ā _g ~(vor). We prove that the extended Torelli map to the perfect cone (1st Voronoi) compactification ā _g ~(perf) is also regular, and moreover ā _g ~(vor) and ā _g ~(perf) share a common Zariski open neighborhood of the image of M? _g. We also show that the map to the Igusa monoidal transform (central cone compactification) is not regular for _g≥9; this disproves a 1973 conjecture of Namikawa.