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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 theDeligne-Mumford compactification $\bar{M}_g$ to the 2nd Voronoicompactification $\bar{A}_g^{vor}$.  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 openneighborhood of the image of $\bar{M}_g$. We also show that the map to theIgusa monoidal transform (central cone compactification) is NOT regular for$g\ge9$; this disproves a 1973 conjecture of Namikawa.

Extending Torelli map to toroidal compactifications of Siegel space