### Extended Torelli Map to the Igusa Blowup in Genus 6, 7, and 8

Valery Alexeev, Ryan Livingston, Joseph Tenini, Maxim Arap, Xiaoyan Hu, Lauren Huckaba, Patrick McFaddin, Stacy Musgrave, Jaeho Shin, and Catherine Ulrich
Source: Experiment. Math. Volume 21, Issue 2 (2012), 193-203.

#### Abstract

It was conjectured in Yukihiko Namikawa, “On the Canonical Holomorphic Map from the Moduli Space of Stable Curves to the Igusa Monoidal Transform,” that the Torelli map $M_g \to A_g$ associating to a curve its Jacobian extends to a regular map from the Deligne–Mumford moduli space of stable curves $\bar{M}_g$ to the (normalization of the) Igusa blowup $\bar{A}^{\rm cent}_g$. A counterexample in genus $g = 9$ was found in Valery Alexeev and Adrian Brunyate, “Extending Torelli Map to Toroidal Compactifications of Siegel Space.” Here, we prove that the extended map is regular for all $g \le 8$, thus completely solving the problem in every genus.

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Primary Subjects: 14D22, 14H10, 14K10, 14M25, 05C75, 11E12, 52C22