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arXiv · 1102.3425

Extending Torelli map to toroidal compactifications of Siegel space

Valery Alexeev·Adrian Brunyate
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Abstract

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.

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Valery Alexeev, Adrian Brunyate. 2011-10-10. Extending Torelli map to toroidal compactifications of Siegel space. https://doi.org/10.1007/s00222-011-0347-2

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