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; Catherine Ulrich

arXiv · 1105.4384

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·Catherine Ulrich

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Abstract

It was conjectured in \cite{Namikawa_ExtendedTorelli} 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 $\bar{A}_g^{\rm cent}$. A counterexample in genus $g=9$ was found in \cite{AlexeevBrunyate}. Here, we prove that the extended map is regular for all $g\le8$, thus completely solving the problem in every genus.

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Valery Alexeev, Ryan Livingston, Joseph Tenini, Maxim Arap, Xiaoyan Hu, Lauren Huckaba, Patrick Mcfaddin, Stacy Musgrave, Jaeho Shin, Catherine Ulrich. 2011-05-23. Extended Torelli map to the Igusa blowup in genus 6, 7, and 8. https://arxiv.org/abs/1105.4384

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