Extending the Torelli map to toroidal compactifications of Siegel space

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