  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.
