Extending the Torelli map to toroidal compactifications of Siegel space

Abstract

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 M g to the 2nd Voronoi compactification A vor g . We prove that the extended Torelli map to the perfect cone (1st Voronoi) compactification A perf g is also regular, and moreover A vor g and A perf g share a common Zariski open neighborhood of the image of 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.

Extending the Prym map to toroidal compactifications of the moduli space of abelian varieties (with an appendix by Mathieu Dutour Sikirić)

“…Assigning to a curve its principally polarized Jacobian defines the Torelli period map M g → A g from the coarse moduli space of curves of genus g to the coarse moduli space of principally polarized abelian varieties (ppav) of dimension g. It is a well known fact, due to Mumford and Namikawa [Nam80], that the Torelli map extends to a morphism M g →Ā V g from the Deligne-Mumford compactification to the second Voronoi toroidal compactification. More recently, Alexeev and Brunyate [AB12] have studied extensions of the Torelli map to other toroidal compactifications and have shown that the period map extends to a morphism to the perfect cone compactificationĀ P g , but not to a morphism to the central cone compactificationĀ C g for g ≥ 9, disproving a conjecture of Namikawa. While the Torelli map is injective for all g, for g ≥ 4 it is not dominant.…”

The Topology of A g $${{{\mathcal A}}_{g}}$$ and Its Compactifications

“…A more interesting case that satisfies the assumptions of the theorem is the matroidal partial compactification A Matr g defined by the fan Σ Matr of cones defined starting from simple regular matroids. This partial compactification was investigated by Melo and Viviani [80] who showed that the matroidal fan with coincides the intersection Σ Matr [4].…”

“…The discrete subgroup Γ g = Sp(2g, Z) will be of special importance for us. The group of (complex) symplectic similitudes is defined by (4) GSp(2g, C) = {M ∈ GL(2g, C) | t M JM = cJ for some c ∈ C * }.…”

Moduli of Weighted Hyperplane Arrangements

“…Gwena [25] gave an example of degenerations of Prym varieties, the intermediate Jacobians of cubic 3-folds, which is described by a dicing for the matroid R 10 . In particular, these Prym varieties are not Jacobians.…”

“…Alexeev and Brunyate showed in [10] that the Torelli map also extends to a morphism M g → A perf g to another interesting compactification, for the perfect fan τ perf . Extending this result, Melo and Viviani [46] proved that the maximal open subset U which is shared by A vor g and A perf g (which are birationally isomorphic, as they both contain A g ) is precisely the locus of dicings corresponding to all regular matroids of rank up to g. The compactified Torelli map factors through U .…”