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 $overline {operatorname {M}}_{g}$ 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.