Extending the Torelli map to toroidal compactifications of Siegel space

Abstract

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 $\overline {\operatorname {M}}_{g}$ to the 2nd Voronoi compactification $\overline {\operatorname {A}}_{g}^{\mathrm {vor}}$. We prove that the extended Torelli map to the perfect cone (1st Voronoi) compactification $\overline {\operatorname {A}}_{g}^{\mathrm {perf}}$ is also regular, and moreover $\overline {\operatorname {A}}_{g}^{\mathrm {vor}}$ and $\overline {\operatorname {A}}_{g}^{\mathrm {perf}}$ share a common Zariski open neighborhood of the image of $\overline {\operatorname {M}}_{g}$. 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.